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Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 1530
Exercise 1Study the shape properties of the f(x)=1 - 40 x3 + 15 x4 + 12 x5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t2+ 2 sin(log(t + 1)) and compute its value at the point t=0.
1) f'(0)=-4 2) f'(0)=1 3) f'(0)=2 4) f'(0)=-1
Exercise 3
Compute the limit: limx→1
11
3- 6 x + 3 x2 - 2 x3
3+ Logx2
1 - 4 x + 6 x2 - 4 x3 + x4
1) 1
2) -1
2
3) -1
4) -∞
5)1
2
6) 0
7) ∞
Exercise 4
Compute the limit: limx→-2-8 - 4 x + 2 x2 + x3
-12 - 8 x + x2 + x3
1) ∞
2) 1
3)4
5
4) -∞
5) -1
6) -2
7) 0
Exercise 5Between the months t=4 and t=9
, the funds in certain account (in millions of euros) are given by the function F(t)=
10 + 120 t - 27 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=4 and t=6.
1) It oscillates between 186 and 197.
2) It oscillates between 185 and 186.
3) It oscillates between 185 and 190.
4) It oscillates between 185 and 361.
5) It oscillates between 188 and 200.
2
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of49 x
9 + 19 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)8
19
2)8
3
3)12
19
4)35
12
5) 2
Exercise 7The yield of a tree plantation is given by f(x)=
-21 + 3 x + 26 x2
14 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)8
7
2) 10
3)31
2
4) 14
5)6
5
Exercise 8Study the differentiability of the function f(x)=
-sin(2 - x) - 2 cos(2 - x) + 3 x ≤ 2ⅇx-2 + 2 cos(2 - x) - 2 2 < x < 4
2 sin(4 - x) - 2 cos(4 - x) + ⅇ2 + 2 cos(2) 4 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=2.
4) The function is differentiable for all points except for x=4.
5) The function is differentiable for all points except for x=2 and x=4.
3
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 339599
Exercise 1Study the shape properties of f(x)=1 - 20 x4 + 2 x6
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=cost3 and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=0 3) f'(0)=1 4) f'(0)=4
Exercise 3
Compute the limit: limx→0
-1 + ⅇx - x -x2
2-
x3
6
x4
1) -1
3
2) 1
3) -2
4) -∞
5) 0
6)1
24
7) ∞
4
Exercise 4
Compute the limit: limx→1-3 + 8 x - 6 x2 + x4
x - 2 x2 + x3
1) -∞
2) -1
2
3) 1
4) 0
5) -1
6) ∞
7) -2
3
Exercise 5Between the months t=1 and t=6
, the true value of the shares of a company (in euros) are given by the function C(t)=
59 + 30 t - 18 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=3 and t=5.
1) It oscillates between 9 and 73.
2) It oscillates between 9 and 73.
3) It oscillates between 0 and 42.
4) It oscillates between 15 and 42.
5) It oscillates between 9 and 41.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of32 x
2 + 22 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)7
11
2)3
11
3) 25
4)11
8
5)17
5
5
Exercise 7The yield of a tree plantation is given by f(x)=
-40 + 15 x + 8 x2
35 x7, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1) 1
2)14
11
3) 23
4)7
4
5) 9
Exercise 8Study the differentiability of the function f(x)=
-2 ⅇx-1 + sin(1) sin(x) + cos(1) cos(x) + 5 x ≤ 12 (-x + cos(1 - x) + 2) 1 < x < 23 cos(2 - x) - 2 + 2 cos(1) 2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=1.
4) The function is differentiable for all points except for x=2.
5) The function is differentiable for all points except for x=1 and x=2.
6
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 453722
Exercise 1Study the shape properties of the f(x)=5 - 6 x2 + 2 x3
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t2- t
3 logsin(t) + 1 and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=-4 3) f'(0)=2 4) f'(0)=0
Exercise 3
Compute the limit: limx→11 - x + Log[x]
1 - 2 x + x2
1) -2
2) ∞
3) -∞
4) -1
5) -1
2
6) 0
7) 1
7
Exercise 4
Compute the limit: limx→1x - 2 x2 + x3
-2 + x + x2
1) -2
3
2) -1
3) 0
4) -∞
5) 1
6) -2
7) ∞
Exercise 5Between the months t=1 and t=6
, the funds in certain account (in millions of euros) are given by the function F(t)=
-3 + 180 t - 33 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=3 and t=4.
1) It oscillates between 285 and 318.
2) It oscillates between 285 and 313.
3) It oscillates between 146 and 322.
4) It oscillates between 321 and 322.
5) It oscillates between 294 and 317.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of50 x
8 + 33 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)7
10
2)5
2
3) 3
4)37
9
5)4
11
8
Exercise 7The yield of a tree plantation is given by f(x)=
-11 + 48 x + 29 x2
x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)7
9
2) 5
3)11
24
4)29
12
5)1
2
Exercise 8Study the differentiability of the function f(x)=
2 sin(x) + cos(x) - 1 x ≤ 0
2 sin(x) - cos(x) + x -1 + sin(3) + 2 cos(3) + 1 0 < x < 3
sin(3 - x) + 3 cos(3 - x) - 5 - sin(3) - 7 cos(3) 3 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=0.
4) The function is differentiable for all points except for x=3.
5) The function is differentiable for all points except for x=0 and x=3.
9
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 4829387
Exercise 1
Study the shape properties of f(x)=3 - 3 x2 +x4
2
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
t + 3 cos(log(t + 1)) + 2 and compute its value at the point t=0.
1) f'(0)=2 2) f'(0)=-1 3) f'(0)=1 4) f'(0)=3
10
Exercise 3
Compute the limit: limx→1
11
2- 9 x +
9 x2
2- x3 + Logx3
1 - 4 x + 6 x2 - 4 x3 + x4
1) ∞
2) -∞
3) 0
4) -2
5) 1
6) -3
4
7) -1
Exercise 4
Compute the limit: limx→-354 + 81 x + 45 x2 + 11 x3 + x4
-9 + 3 x + 5 x2 + x3
1) ∞
2) -1
2
3) -1
4) 1
5) 0
6) -∞
7) -2
3
Exercise 5Between the months t=3 and t=7
, the true value of the shares of a company (in euros) are given by the function C(t)=
91 + 108 t - 27 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=3 and t=6.
1) It oscillates between 207 and 229.
2) It oscillates between 206 and 221.
3) It oscillates between 199 and 226.
4) It oscillates between 197 and 225.
5) It oscillates between 209 and 230.
11
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of18 x
8 + 33 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1) 9
2)11
10
3)4
33
4) 4
5) 4
Exercise 7The yield of a tree plantation is given by f(x)=
-26 + 11 x + 20 x2
38 x3, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)23
5
2)6
7
3)38
5
4)3
2
5)15
7
Exercise 8
Study the differentiability of the function f(x)=
-ⅇx+3 - 3 cos(x + 3) - 3 x ≤ -31
4(x (x + 2) - 31) -3 < x < -1
x - (x + 2) log(x + 2) - 7 -1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-3.
4) The function is differentiable for all points except for x=-1.
5) The function is differentiable for all points except for x=-3 and x=-1.
12
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 7404309
Exercise 1Study the shape properties of the f(x)=1 - 3 x2 + 2 x3
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t + sin(t) + ⅇ and compute its value at the point t=0.
1) f'(0)=1 2) f'(0)=-4 3) f'(0)=-3 4) f'(0)=2
Exercise 3
Compute the limit: limx→11 - x + Log[x]
1 - 2 x + x2
1) 1
2) -1
3
3) ∞
4) -2
5) -1
2
6) 0
7) -∞
13
Exercise 4
Compute the limit: limx→1-1 + 2 x - 2 x3 + x4
x - 2 x2 + x3
1) -∞
2) 0
3) -2
4) 1
5) -2
3
6) -1
7) ∞
Exercise 5Between the months t=3 and t=9
, the funds in certain account (in millions of euros) are given by the function F(t)=
-13 + 144 t - 33 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=3 and t=4.
1) It oscillates between 161 and 182.
2) It oscillates between 51 and 176.
3) It oscillates between 51 and 176.
4) It oscillates between 171 and 168.
5) It oscillates between 163 and 176.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of16 x
4 + 4 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)4
5
2) 1
3) 12
4)29
20
5)29
17
14
Exercise 7The yield of a tree plantation is given by f(x)=
-32 + 22 x + 25 x2
37 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1) 1
2)37
18
3)29
12
4)24
11
5)32
11
Exercise 8
Study the differentiability of the function f(x)=
-cos(x + 1) x ≤ -11
2(x (x + 2) - 1) -1 < x < 0
-sin(x) + 3 cos(x) -7
20 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=0.
5) The function is differentiable for all points except for x=-1 and x=0.
15
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 7511795
Exercise 1Study the shape properties of the f(x)=1 - 24 x - 18 x2 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
-log(t + 1) and compute its value at the point t=0.
1) f'(0)=-1 2) f'(0)=3 3) f'(0)=4 4) f'(0)=2
16
Exercise 3
Compute the limit: limx→0-x + Sin[x]
x3
1) -1
6
2) 1
3) 0
4) ∞
5) -1
6) -∞
7) -2
3
Exercise 4
Compute the limit: limx→1-3 + 7 x - 5 x2 + x3
2 - 3 x + x2
1) 0
2) -1
3) ∞
4) -∞
5) 1
6) -1
2
7) -2
Exercise 5Between the months t=4 and t=11
, the funds in certain account (in millions of euros) are given by the function F(t)=
-13 + 240 t - 42 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=5 and t=10.
1) It oscillates between 187 and 403.
2) It oscillates between 180 and 390.
3) It oscillates between 187 and 403.
4) It oscillates between 183 and 388.
5) It oscillates between 187 and 387.
17
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of49 x
4 + 40 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)10
7
2)3
4
3) 1
4)1
4
5)5
16
Exercise 7The yield of a tree plantation is given by f(x)=
-20 + 34 x + 49 x2
7 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)13
8
2)15
4
3)20
17
4)17
2
5)40
3
Exercise 8Study the differentiability of the function f(x)=
-sin(x) + 3 cos(x) + 4 x ≤ 0x - 2 sin(x) - cos(x) + 8 0 < x < 1ⅇx-1 - cos(1 - x) + 12 - 2 sin(1) - cos(1) 1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=0.
4) The function is differentiable for all points except for x=1.
5) The function is differentiable for all points except for x=0 and x=1.
18
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 7572959
Exercise 1Study the shape properties of the f(x)=3 + 12 x2 - 12 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
-2 ⅇt+ cos(t) + 3 and compute its value at the point t=0.
1) f'(0)=4 2) f'(0)=0 3) f'(0)=-2 4) f'(0)=-4
19
Exercise 3
Compute the limit: limx→13 - 4 x + x2 + Logx2
-1 + 3 x - 3 x2 + x3
1) 0
2) ∞
3) -2
4) -1
2
5)2
3
6) -∞
7) 1
Exercise 4
Compute the limit: limx→26 - 5 x + x2
-6 + x + x2
1) ∞
2) 0
3) 1
4) -1
5) -2
6) -1
5
7) -∞
Exercise 5Between the months t=0 and t=4
, the funds in certain account (in millions of euros) are given by the function F(t)=
36 t - 15 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=1 and t=3.
1) It oscillates between 23 and 22.
2) It oscillates between 23 and 28.
3) It oscillates between 33 and 24.
4) It oscillates between 27 and 28.
5) It oscillates between 0 and 32.
20
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of36 x
1 + 24 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)5
24
2) 25
3) 2
4)23
17
5)17
12
Exercise 7The yield of a tree plantation is given by f(x)=
-49 + 38 x + 19 x2
9 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1) 1
2)3
17
3)9
7
4)28
11
5)49
19
Exercise 8
Study the differentiability of the function f(x)=2 sin(3 - x) + 3 cos(3 - x) x ≤ 39 - 2 x 3 < x < 51 - 2 cos(5 - x) 5 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=5.
5) The function is differentiable for all points except for x=3 and x=5.
21
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 7684103
Exercise 1Study the shape properties of the f(x)=2 - 24 x2 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
log(t + 1) log(t + 1) - 2 ⅇt cos(t) and compute its value at the point t=0.
1) f'(0)=3 2) f'(0)=4 3) f'(0)=-2 4) f'(0)=-3
22
Exercise 3
Compute the limit: limx→13 - 4 x + x2 + Logx2
-1 + 3 x - 3 x2 + x3
1) ∞
2) -1
3
3)2
3
4) 0
5) -∞
6) 1
7) -2
3
Exercise 4
Compute the limit: limx→24 - 3 x2 + x3
-6 + x + x2
1) -1
2) 1
3) ∞
4) 0
5) -2
3
6) -2
7) -∞
Exercise 5Between the months t=1 and t=5
, the funds in certain account (in millions of euros) are given by the function F(t)=
7 + 12 t - 9 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=1 and t=2.
1) It oscillates between 20 and 12.
2) It oscillates between 20 and 6.
3) It oscillates between 11 and 12.
4) It oscillates between 11 and 92.
5) It oscillates between 6 and 14.
23
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of36 x
1 + 22 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)5
22
2)29
5
3)5
2
4)19
7
5) 2
Exercise 7The yield of a tree plantation is given by f(x)=
-39 + 47 x + 44 x2
30 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)3
13
2)28
5
3)38
5
4)78
47
5)18
19
Exercise 8Study the differentiability of the function f(x)=
2 sin(3 - x) - cos(3 - x) - 3 x ≤ 3-4 x + 2 ⅇx-3 - cos(3 - x) + 7 3 < x < 6
-(x - 5) log(x - 5) + 2 ⅇ3 - 17 - cos(3) 6 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=6.
5) The function is differentiable for all points except for x=3 and x=6.
24
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 8463511
Exercise 1Study the shape properties of f(x)=1 - 5 x4 + 2 x6
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
t2(-2 log(t + 1) + cos(t) + 1) and compute its value at the point t=0.
1) f'(0)=4 2) f'(0)=-4 3) f'(0)=-2 4) f'(0)=0
25
Exercise 3
Compute the limit: limx→0-1 + Cosx2
x3
1) 0
2) -1
3) ∞
4) 1
5) -1
3
6)1
3
7) -∞
Exercise 4
Compute the limit: limx→1-3 + 2 x + x2
2 - 3 x + x2
1) -∞
2) 1
3) -1
4) 0
5) -2
3
6) ∞
7) -4
Exercise 5Between the months t=1 and t=7
, the true value of the shares of a company (in euros) are given by the function C(t)=
7 + 12 t - 9 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=2 and t=7.
1) It oscillates between 11 and 336.
2) It oscillates between 10 and 327.
3) It oscillates between 11 and 12.
4) It oscillates between 7 and 334.
5) It oscillates between 1 and 335.
26
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of50 x
8 + 22 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)31
14
2) 1
3) 12
4)6
11
5)13
2
Exercise 7The yield of a tree plantation is given by f(x)=
-16 + 10 x + 33 x2
31 x8, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)1
13
2)64
99
3) 8
4)25
3
5)22
9
Exercise 8Study the differentiability of the function f(x)=
ⅇx-2 - 3 sin(2) sin(x) - 3 cos(2) cos(x) - 2 x ≤ 2
-x2 + 5 x - 10 2 < x < 3
sin(3 - x) + cos(3 - x) - 1 3 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=2.
4) The function is differentiable for all points except for x=3.
5) The function is differentiable for all points except for x=2 and x=3.
27
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 26522947
Exercise 1
Study the shape properties of f(x)=4 - 12 x2 - 6 x3 +3 x5
5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
sin(t) - 3 ⅇt logsin(t) + 1 and compute its value at the point t=0.
1) f'(0)=0 2) f'(0)=1 3) f'(0)=-1 4) f'(0)=-2
28
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x5
1) 0
2) ∞
3) 1
4) -2
5)1
2
6) -∞
7) -1
3
Exercise 4
Compute the limit: limx→39 x - 6 x2 + x3
-6 - x + x2
1) 1
2) -1
3) ∞
4) 0
5) -2
6) -2
3
7) -∞
Exercise 5Between the months t=1 and t=6
, the true value of the shares of a company (in euros) are given by the function C(t)=
26 + 24 t - 15 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=3 and t=6.
1) It oscillates between 10 and 59.
2) It oscillates between 2 and 69.
3) It oscillates between 10 and 37.
4) It oscillates between 16 and 60.
5) It oscillates between 10 and 62.
29
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of9 x
1 + 20 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)24
11
2)1
10
3)31
15
4)22
7
5)30
13
Exercise 7The yield of a tree plantation is given by f(x)=
-24 + 14 x + 44 x2
47 x6, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1) 6
2)8
15
3)8
11
4)7
10
5)19
5
Exercise 8
Study the differentiability of the function f(x)=
2 ⅇx - 2 cos(x) + 5 x ≤ 0
-3 x2
2+ 4 x + 5 0 < x < 2
9 - 2 ⅇx-2 2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=0.
4) The function is differentiable for all points except for x=2.
5) The function is differentiable for all points except for x=0 and x=2.
30
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 45942139
Exercise 1Study the shape properties of the f(x)=4 - 24 x + 30 x2 - 16 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t2+ 2 sin2(t) and compute its value at the point t=0.
1) f'(0)=1 2) f'(0)=0 3) f'(0)=2 4) f'(0)=-1
Exercise 3
Compute the limit: limx→1
11
6- 3 x +
3 x2
2-
x3
3+ Log[x]
1 - 4 x + 6 x2 - 4 x3 + x4
1) -∞
2) -2
3) ∞
4) 0
5) -1
6) 1
7) -1
4
31
Exercise 4
Compute the limit: limx→-1-2 - 3 x + x3
-1 - x + x2 + x3
1)3
2
2) 1
3) -2
3
4) 0
5) -1
6) -∞
7) ∞
Exercise 5Between the months t=3 and t=9
, the funds in certain account (in millions of euros) are given by the function F(t)=
-11 + 288 t - 42 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=3 and t=8.
1) It oscillates between 526 and 629.
2) It oscillates between 629 and 637.
3) It oscillates between 525 and 641.
4) It oscillates between 529 and 635.
5) It oscillates between 529 and 637.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of25 x
1 + 39 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)7
16
2)4
39
3)11
2
4)27
10
5)4
5
32
Exercise 7The yield of a tree plantation is given by f(x)=
-37 + 41 x + 28 x2
44 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)74
41
2) 4
3)4
5
4)38
11
5)8
9
Exercise 8
Study the differentiability of the function f(x)=
2 sin(x + 2) + 2 cos(x + 2) + 5 x ≤ -2
-x2
2+ x + 11 -2 < x < -1
2 ⅇx+1 - cos(x + 1) +17
2-1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-2.
4) The function is differentiable for all points except for x=-1.
5) The function is differentiable for all points except for x=-2 and x=-1.
33
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 46272561
Exercise 1Study the shape properties of f(x)=4 + 80 x3 - 20 x4 - 6 x5 + 2 x6
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
cos(t) log(t + 1) sin(t) + 1 and compute its value at the point t=0.
1) f'(0)=-1 2) f'(0)=-2 3) f'(0)=-4 4) f'(0)=0
Exercise 3
Compute the limit: limx→0
-1 + ⅇx - x -x2
2
x3
1) -∞
2) ∞
3) 1
4) -2
5)1
6
6) 0
7) -1
34
Exercise 4
Compute the limit: limx→1-2 + 5 x - 4 x2 + x3
1 - x - x2 + x3
1) 0
2) 1
3) ∞
4) -2
5) -1
6) -∞
7) -1
2
Exercise 5Between the months t=1 and t=6
, the true value of the shares of a company (in euros) are given by the function C(t)=
115 + 108 t - 27 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=2 and t=4.
1) It oscillates between 223 and 250.
2) It oscillates between 239 and 252.
3) It oscillates between 237 and 250.
4) It oscillates between 198 and 250.
5) It oscillates between 239 and 250.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of36 x
9 + 9 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1) 1
2)19
4
3)35
8
4)7
3
5)22
19
35
Exercise 7The yield of a tree plantation is given by f(x)=
-14 + 39 x + 36 x2
35 x7, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)1
3
2)34
3
3)36
13
4)22
13
5) 21
Exercise 8
Study the differentiability of the function f(x)=
ⅇx-1 + sin(1) sin(x) + cos(1) cos(x) x ≤ 11
2x2 + 3 1 < x < 4
-2 sin(4 - x) + cos(4 - x) +17
24 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=1.
4) The function is differentiable for all points except for x=4.
5) The function is differentiable for all points except for x=1 and x=4.
36
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77024362
Exercise 1
Study the shape properties of f(x)=4 + 2 x3 +x4
2
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
3 (log(t + 1) + 1) + sin(log(t + 1)) and compute its value at the point t=0.
1) f'(0)=4 2) f'(0)=-4 3) f'(0)=0 4) f'(0)=2
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x4
1) 0
2) ⅇ8
3) ∞
4) 1
5) 2
6) -2
7) -∞
37
Exercise 4
Compute the limit: limx→3-18 + 21 x - 8 x2 + x3
9 + 3 x - 5 x2 + x3
1) ∞
2) -2
3) -∞
4)1
4
5) -1
6) 1
7) 0
Exercise 5Between the months t=2 and t=9
, the true value of the shares of a company (in euros) are given by the function C(t)=
556 + 378 t - 48 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=3 and t=4.
1) It oscillates between 1528 and 1536.
2) It oscillates between 1312 and 1428.
3) It oscillates between 1318 and 1420.
4) It oscillates between 1136 and 1536.
5) It oscillates between 1308 and 1423.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of27 x
12 + 20 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)3
10
2)17
2
3) 1
4) 3
5)38
11
38
Exercise 7The yield of a tree plantation is given by f(x)=
-49 + 21 x + 27 x2
5 x5, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)35
27
2)25
11
3)17
2
4)23
13
5)1
6
Exercise 8Study the differentiability of the function f(x)=
2 ⅇx+1 - cos(x + 1) + 1 x ≤ -1
4 x - 2 ⅇx+1 + cos(x + 1) + 7 -1 < x < 2
2 ⅇx-2 + 2 cos(2 - x) - 2 ⅇ3 + 11 + cos(3) 2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=2.
5) The function is differentiable for all points except for x=-1 and x=2.
39
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77352873
Exercise 1
Study the shape properties of f(x)=4 + 6 x2 + 3 x3 +x4
2
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
sinsin(t) + 3 logsin3(t) + 1 and compute its value at the point t=0.
1) f'(0)=1 2) f'(0)=-3 3) f'(0)=-2 4) f'(0)=-4
40
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x6
1)1
2
2) -1
3) -∞
4) 0
5) 1
6) -2
7) ∞
Exercise 4
Compute the limit: limx→354 - 81 x + 45 x2 - 11 x3 + x4
-9 + 15 x - 7 x2 + x3
1) 1
2) 0
3) ∞
4) -∞
5) -2
6) -1
7) -2
3
Exercise 5Between the months t=4 and t=10
, the true value of the shares of a company (in euros) are given by the function C(t)=
215 + 168 t - 33 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=8 and t=10.
1) It oscillates between 460 and 487.
2) It oscillates between 460 and 595.
3) It oscillates between 480 and 599.
4) It oscillates between 477 and 598.
5) It oscillates between 471 and 595.
41
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of48 x
12 + 7 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)38
15
2)37
4
3)5
4
4)12
7
5)1
2
Exercise 7The yield of a tree plantation is given by f(x)=
-6 + 28 x + 50 x2
33 x6, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)1
9
2)2
19
3)13
6
4)10
3
5)1
5
Exercise 8Study the differentiability of the function f(x)=
-ⅇx - 2 cos(x) + 4 x ≤ 0
x - 3 x sin(1) - 2 sin(x) + 2 x cos(1) - 3 cos(x) + 6 0 < x < 1-sin(1 - x) + 3 cos(1 - x) + 4 - 5 sin(1) - cos(1) 1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=0.
4) The function is differentiable for all points except for x=1.
5) The function is differentiable for all points except for x=0 and x=1.
42
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77361996
Exercise 1Study the shape properties of the f(x)=5 - 96 x - 24 x2 + 8 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
ⅇt+ sinsin(t) + ⅇ
sin(t) and compute its value at the point t=0.
1) f'(0)=-4 2) f'(0)=3 3) f'(0)=0 4) f'(0)=-2
Exercise 3
Compute the limit: limx→0
-1 + ⅇx - x -x2
2
x3
1) -∞
2) 1
3) ∞
4) 0
5) -1
6) -2
7)1
6
43
Exercise 4
Compute the limit: limx→12 - 3 x + x3
3 - 5 x + x2 + x3
1) -∞
2) 0
3) -2
4) -1
5)3
4
6) 1
7) ∞
Exercise 5Between the months t=0 and t=4
, the funds in certain account (in millions of euros) are given by the function F(t)=
19 + 18 t - 12 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=3 and t=4.
1) It oscillates between 25 and 34.
2) It oscillates between 11 and 33.
3) It oscillates between 19 and 27.
4) It oscillates between 18 and 35.
5) It oscillates between 15 and 28.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of48 x
12 + 20 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)8
9
2)9
7
3)3
5
4) 1
5) 2
44
Exercise 7The yield of a tree plantation is given by f(x)=
-45 + 40 x + 5 x2
22 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)17
13
2)9
4
3)23
16
4) 2
5) 8
Exercise 8
Study the differentiability of the function f(x)=
5 - 2 sin(x + 1) x ≤ -13 x2
2+ x +
9
2-1 < x < 1
7 - 2 sin(1 - x) 1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=1.
5) The function is differentiable for all points except for x=-1 and x=1.
45
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77377313
Exercise 1
Study the shape properties of f(x)=2 + 12 x2 + 10 x3 + 4 x4 +3 x5
5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
sint3 - 3 cost3 + 2 and compute its value at the point t=0.
1) f'(0)=0 2) f'(0)=-1 3) f'(0)=2 4) f'(0)=-2
46
Exercise 3
Compute the limit: limx→0-1 + Cosx3
x5
1) 1
2) ∞
3) -∞
4) 0
5)1
3
6) -1
2
7)2
3
Exercise 4
Compute the limit: limx→-33 + 4 x + x2
6 + 5 x + x2
1) -∞
2) 0
3) -1
4) -2
5) 2
6) 1
7) ∞
Exercise 5Between the months t=0 and t=4
, the true value of the shares of a company (in euros) are given by the function C(t)=
44 + 48 t - 18 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=1 and t=2.
1) It oscillates between 76 and 84.
2) It oscillates between 82 and 91.
3) It oscillates between 44 and 84.
4) It oscillates between 76 and 75.
5) It oscillates between 70 and 90.
47
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of36 x
1 + 9 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)23
20
2)15
11
3)5
9
4)33
5
5)31
15
Exercise 7The yield of a tree plantation is given by f(x)=
-4 + x + 44 x2
47 x3, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)21
20
2)19
3
3)1
2
4)3
11
5) 10
Exercise 8
Study the differentiability of the function f(x)=3 cos(x + 3) x ≤ -3x + 6 -3 < x < -1ⅇx+1 + 4 -1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-3.
4) The function is differentiable for all points except for x=-1.
5) The function is differentiable for all points except for x=-3 and x=-1.
48
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77380424
Exercise 1
Study the shape properties of f(x)=1 - 12 x2 - 6 x3 +3 x5
5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
sin(t) + 1 - 3 ⅇt cos(t) and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=4 3) f'(0)=-3 4) f'(0)=2
49
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x6
1) ∞
2) 0
3) 1
4)1
2
5) -2
6) -∞
7) -1
Exercise 4
Compute the limit: limx→36 - 5 x + x2
-9 + x2
1) -1
2)1
6
3) 0
4) -∞
5) -2
3
6) 1
7) ∞
Exercise 5Between the months t=0 and t=6
, the true value of the shares of a company (in euros) are given by the function C(t)=
10 + 48 t - 18 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=1 and t=5.
1) It oscillates between 48 and 57.
2) It oscillates between 35 and 56.
3) It oscillates between 50 and 45.
4) It oscillates between 42 and 50.
5) It oscillates between 10 and 82.
50
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of25 x
16 + 31 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)4
31
2)19
10
3)2
7
4)5
3
5)33
20
Exercise 7The yield of a tree plantation is given by f(x)=
-25 + 50 x + 15 x2
29 x5, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)5
9
2)37
16
3)8
5
4)33
20
5)11
5
Exercise 8Study the differentiability of the function f(x)=
2 cos(x + 1) - 3 x ≤ -1-2 x + 2 sin(x + 1) - 3 cos(x + 1) -1 < x < 22 sin(2 - x) - 4 + 2 sin(3) - 3 cos(3) 2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=2.
5) The function is differentiable for all points except for x=-1 and x=2.
51
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77383294
Exercise 1Study the shape properties of the f(x)=5 + 24 x - 6 x2 - 8 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
ⅇt+ 2 sin(t) cos2(t) and compute its value at the point t=0.
1) f'(0)=3 2) f'(0)=-3 3) f'(0)=-1 4) f'(0)=-4
Exercise 3
Compute the limit: limx→13 - 4 x + x2 + Logx2
-1 + 3 x - 3 x2 + x3
1) ∞
2)2
3
3) 1
4) -∞
5) 0
6) -2
7) -1
52
Exercise 4
Compute the limit: limx→-224 + 44 x + 30 x2 + 9 x3 + x4
-12 - 8 x + x2 + x3
1) -1
2) -1
2
3) 0
4) -2
5) ∞
6) 1
7) -∞
Exercise 5Between the months t=0 and t=5
, the funds in certain account (in millions of euros) are given by the function F(t)=
16 - 3 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=0 and t=1.
1) It oscillates between 20 and 21.
2) It oscillates between 19 and 16.
3) It oscillates between 15 and 25.
4) It oscillates between 15 and 16.
5) It oscillates between 15 and 191.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of27 x
3 + 38 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1) 11
2) 1
3)17
8
4) 1
5)3
19
53
Exercise 7The yield of a tree plantation is given by f(x)=
-29 + 36 x + 31 x2
3 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)29
18
2) 13
3)4
19
4)5
13
5)21
20
Exercise 8Study the differentiability of the function f(x)=
3 cos(2 - x) + 2 x ≤ 2
7 - 2 x + sin(2 - x) - cos(2 - x) 2 < x < 3
-4 x + 3 (x - 2) log(x - 2) + 13 + 2 sin(1) + 2 cos(1) 3 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=2.
4) The function is differentiable for all points except for x=3.
5) The function is differentiable for all points except for x=2 and x=3.
54
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77386231
Exercise 1Study the shape properties of the f(x)=2 - 240 x - 240 x2 - 60 x3 + 30 x4 + 12 x5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
logt2 + 1 sint2 - 3 logt2 + 1 and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=4 3) f'(0)=-1 4) f'(0)=0
55
Exercise 3
Compute the limit: limx→0-1 + ⅇx
2
- x2
x3
1) ∞
2) 1
3) 0
4) -2
3
5) -1
2
6) -∞
7) -1
Exercise 4
Compute the limit: limx→2-2 x + x2
-6 + x + x2
1) 1
2) ∞
3) -1
4) -2
5) -∞
6)2
5
7) 0
Exercise 5Between the months t=0 and t=7
, the funds in certain account (in millions of euros) are given by the function F(t)=
1 + 120 t - 27 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=2 and t=4.
1) It oscillates between 144 and 171.
2) It oscillates between 176 and 177.
3) It oscillates between 149 and 177.
4) It oscillates between 1 and 204.
5) It oscillates between 147 and 173.
56
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of48 x
12 + 24 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)28
5
2)1
2
3) 6
4) 16
5)32
11
Exercise 7The yield of a tree plantation is given by f(x)=
-35 + 2 x + 34 x2
34 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)28
5
2) 35
3) 6
4)8
13
5)11
6
Exercise 8Study the differentiability of the function f(x)=
2 sin(3 - x) + 5 x ≤ 3
x log x-2
4 - 2 log(x - 2) + 5 + log(64) 3 < x < 6
-sin(6 - x) + 5 + log(4) 6 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=6.
5) The function is differentiable for all points except for x=3 and x=6.
57
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77387031
Exercise 1
Study the shape properties of f(x)=1 + 6 x2 - 3 x3 +x4
2
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
3 ⅇⅇt
ⅇt+ 1 and compute its value at the point t=0.
1) f'(0)=1 2) f'(0)=9 ⅇ 3) f'(0)=3 4) f'(0)=-2
58
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x6
1) 0
2) -∞
3) -1
4) -2
5)1
2
6) ∞
7) 1
Exercise 4
Compute the limit: limx→354 - 81 x + 45 x2 - 11 x3 + x4
9 + 3 x - 5 x2 + x3
1) ∞
2) -2
3) -∞
4) -1
5) -2
3
6) 1
7) 0
Exercise 5Between the months t=2 and t=7
, the true value of the shares of a company (in euros) are given by the function C(t)=
298 + 210 t - 36 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=4 and t=6.
1) It oscillates between 696 and 689.
2) It oscillates between 590 and 698.
3) It oscillates between 687 and 698.
4) It oscillates between 690 and 698.
5) It oscillates between 690 and 706.
59
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of16 x
1 + 13 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1) 2
2)10
3
3)3
2
4)3
13
5) 22
Exercise 7The yield of a tree plantation is given by f(x)=
-5 + 12 x2
13 x5, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)18
13
2)38
15
3)22
9
4)5
6
5)16
9
Exercise 8Study the differentiability of the function f(x)=
5 - cos(x + 1) x ≤ -1-2 x + 2 ⅇx+1 - 2 ⅇ2 (x + 1) + 3 x sin(2) + 3 cos(x + 1) - 3 + 3 sin(2) -1 < x < 1
x (log(x) - 3) - 2 ⅇ2 - 2 + 6 sin(2) + 3 cos(2) 1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=1.
5) The function is differentiable for all points except for x=-1 and x=1.
60
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77434209
Exercise 1Study the shape properties of the f(x)=2 + 480 x - 120 x2 - 120 x3 + 15 x4 + 12 x5
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t + log(t + 1) - 3 ⅇsin(t) and compute its value at the point t=0.
1) f'(0)=-1 2) f'(0)=0 3) f'(0)=-4 4) f'(0)=3
61
Exercise 3
Compute the limit: limx→0-1 + Cosx2
x3
1) -2
3
2) -2
3) 0
4) ∞
5) 1
6) -1
2
7) -∞
Exercise 4
Compute the limit: limx→2-2 - x + x2
-2 x + x2
1) -2
2) 1
3) 0
4) ∞
5) -∞
6) -1
7)3
2
Exercise 5Between the months t=2 and t=7
, the funds in certain account (in millions of euros) are given by the function F(t)=
-3 + 48 t - 18 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=2 and t=3.
1) It oscillates between 33 and 37.
2) It oscillates between 29 and 137.
3) It oscillates between 29 and 37.
4) It oscillates between 34 and 39.
5) It oscillates between 23 and 43.
62
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of18 x
2 + 45 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)16
3
2)18
5
3)29
6
4)28
11
5)4
45
Exercise 7The yield of a tree plantation is given by f(x)=
-7 + x + 48 x2
40 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)9
10
2) 8
3)17
5
4)15
7
5) 14
Exercise 8Study the differentiability of the function f(x)=
sin(3 - x) - 3 cos(3 - x) + 4 x ≤ 3-2 ⅇ2 (x - 3) + 2 ⅇx-3 + x + x sin(2) + cos(3 - x) - 5 - 3 sin(2) 3 < x < 5
-sin(5 - x) + cos(5 - x) - 2 ⅇ2 - 1 + 2 sin(2) + cos(2) 5 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=5.
5) The function is differentiable for all points except for x=3 and x=5.
63
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77644810
Exercise 1Study the shape properties of the f(x)=5 + 24 x2 + 16 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
sin(t) + 2 ⅇt+ sin(t) and compute its value at the point t=0.
1) f'(0)=4 2) f'(0)=5 3) f'(0)=2 4) f'(0)=-3
64
Exercise 3
Compute the limit: limx→0
-x +x3
6+ Sin[x]
x4
1) -∞
2) -1
2
3) 0
4) -1
3
5) 1
6) ∞
7) -2
Exercise 4
Compute the limit: limx→-1-1 - 2 x + 2 x3 + x4
-2 - 3 x + x3
1) ∞
2) 1
3) 0
4) -∞
5) -2
6) -1
7) -1
2
Exercise 5Between the months t=4 and t=9
, the funds in certain account (in millions of euros) are given by the function F(t)=
-7 + 432 t - 51 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=4 and t=7.
1) It oscillates between 1033 and 1209.
2) It oscillates between 1027 and 1203.
3) It oscillates between 1208 and 1209.
4) It oscillates between 1033 and 1204.
5) It oscillates between 1031 and 1199.
65
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of36 x
25 + 43 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)19
18
2)5
43
3)10
7
4)34
7
5)5
14
Exercise 7The yield of a tree plantation is given by f(x)=
-29 + 46 x + 41 x2
25 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)19
18
2)29
23
3)10
7
4)34
7
5)3
5
Exercise 8
Study the differentiability of the function f(x)=
-2 sin(x + 2) - 3 cos(x + 2) + 1 x ≤ -23 x2
4+ x - 3 -2 < x < 0
ⅇx + 3 cos(x) - 7 0 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-2.
4) The function is differentiable for all points except for x=0.
5) The function is differentiable for all points except for x=-2 and x=0.
66
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77646467
Exercise 1Study the shape properties of the f(x)=1 - 6 x2 + 2 x3
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
logⅇt + 1 + ⅇⅇt cosⅇt and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=0 3) f'(0)=1
2+ ⅇ Cos[1] - ⅇ Sin[1] 4) f'(0)=-1
67
Exercise 3
Compute the limit: limx→0
-1 + ⅇx - x -x2
2-
x3
6
x4
1) -2
2) 1
3) -2
3
4)1
24
5) -∞
6) 0
7) ∞
Exercise 4
Compute the limit: limx→1-3 + 8 x - 6 x2 + x4
2 - 3 x + x3
1) 1
2) ∞
3) -2
3
4) -1
5) -1
2
6) -∞
7) 0
Exercise 5Between the months t=1 and t=5
, the funds in certain account (in millions of euros) are given by the function F(t)=
-1 + 12 t - 9 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=1 and t=4.
1) It oscillates between 3 and 4.
2) It oscillates between 3 and 84.
3) It oscillates between -3 and 27.
4) It oscillates between 7 and 21.
5) It oscillates between 3 and 31.
68
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of45 x
5 + xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)5
19
2)19
3
3)31
2
4) 10
5)21
5
Exercise 7The yield of a tree plantation is given by f(x)=
-42 + 12 x + 20 x2
14 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)37
16
2) 7
3) 1
4) 19
5)2
3
Exercise 8Study the differentiability of the function f(x)=
3 cos(x + 1) + 5 x ≤ -1x (3 + log(4)) - (x + 2) log(x + 2) + 11 + log(4) -1 < x < 2-2 sin(2 - x) - 3 cos(2 - x) + 22 - log(4) 2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=2.
5) The function is differentiable for all points except for x=-1 and x=2.
69
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77688139
Exercise 1Study the shape properties of the f(x)=4 - 48 x - 24 x2 + 4 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
logt2 + 1 - 2 logt2 + 1 sint2 and compute its value at the point t=0.
1) f'(0)=-4 2) f'(0)=-3 3) f'(0)=-1 4) f'(0)=0
70
Exercise 3
Compute the limit: limx→0-1 + Cosx2
x3
1) -2
3
2) 0
3) 1
4) ∞
5) -1
2
6) -∞
7)1
3
Exercise 4
Compute the limit: limx→2-2 x + x2
-2 - x + x2
1) 0
2) -∞
3) -2
3
4)2
3
5) -1
6) ∞
7) 1
Exercise 5Between the months t=4 and t=11
, the funds in certain account (in millions of euros) are given by the function F(t)=
-7 + 288 t - 42 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=5 and t=6.
1) It oscillates between 633 and 641.
2) It oscillates between 601 and 741.
3) It oscillates between 623 and 632.
4) It oscillates between 631 and 637.
5) It oscillates between 634 and 647.
71
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of49 x
1 + 2 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)16
5
2)7
3
3) 3
4)23
20
5)2
11
Exercise 7The yield of a tree plantation is given by f(x)=
-32 + 9 x + 42 x2
3 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)64
9
2) 11
3)40
9
4)40
3
5)14
9
Exercise 8Study the differentiability of the function f(x)=
-2 sin(3 - x) + 3 cos(3 - x) - 5 x ≤ 32 (x - 2) log(x - 2) - 2 3 < x < 6-x + 3 (x - 5) log(x - 5) + 4 + 8 log(4) 6 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=6.
5) The function is differentiable for all points except for x=3 and x=6.
72
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77770524
Exercise 1Study the shape properties of the f(x)=3 - 24 x - 6 x2 + 8 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t3+ ⅇ
t- 2 ⅇ
t cos(t) and compute its value at the point t=0.
1) f'(0)=-1 2) f'(0)=-3 3) f'(0)=-2 4) f'(0)=3
Exercise 3
Compute the limit: limx→0-1 + Cosx2
x4
1) 1
2) ∞
3) -2
4) -1
5) -∞
6) -1
2
7) 0
73
Exercise 4
Compute the limit: limx→2-6 + x + x2
-4 + x2
1) -1
2) -2
3) 1
4) 0
5) ∞
6) -∞
7)5
4
Exercise 5Between the months t=4 and t=8
, the funds in certain account (in millions of euros) are given by the function F(t)=
-17 + 180 t - 33 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=4 and t=7.
1) It oscillates between 303 and 312.
2) It oscillates between 309 and 309.
3) It oscillates between 304 and 309.
4) It oscillates between 307 and 308.
5) It oscillates between 303 and 335.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of25 x
4 + 45 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)37
7
2) 1
3) 26
4)22
13
5)2
15
74
Exercise 7The yield of a tree plantation is given by f(x)=
-7 + 7 x + 3 x2
11 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)25
13
2)1
3
3)1
20
4)37
9
5) 2
Exercise 8
Study the differentiability of the function f(x)=3 cos(3 - x) + 1 x ≤ 3-x - sin(3 - x) + 7 3 < x < 4-ⅇx-4 - cos(4 - x) + 9 + sin(1) 4 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=4.
5) The function is differentiable for all points except for x=3 and x=4.
75
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 77771717
Exercise 1Study the shape properties of f(x)=4 + 60 x2 + 10 x3 - 15 x4 - 3 x5 + 2 x6
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
ⅇt+ 2 log(t + 1) sinⅇt and compute its value at the point t=0.
1) f'(0)=-2 2) f'(0)=1 + 2 Sin[1] 3) f'(0)=0 4) f'(0)=-4
76
Exercise 3
Compute the limit: limx→1
11
3- 6 x + 3 x2 - 2 x3
3+ Logx2
1 - 4 x + 6 x2 - 4 x3 + x4
1) -∞
2)1
2
3) ∞
4) 1
5) -1
3
6) 0
7) -1
2
Exercise 4
Compute the limit: limx→318 - 3 x - 4 x2 + x3
9 x - 6 x2 + x3
1)5
3
2) -2
3) ∞
4) -1
5) 0
6) 1
7) -∞
Exercise 5Between the months t=4 and t=11
, the true value of the shares of a company (in euros) are given by the function C(t)=
742 + 324 t - 45 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=4 and t=11.
1) It oscillates between 1454 and 1523.
2) It oscillates between 1471 and 1498.
3) It oscillates between 1453 and 1518.
4) It oscillates between 1446 and 1523.
5) It oscillates between 1446 and 1533.
77
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of20 x
5 + 45 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1) 2
2) 3
3) 7
4)19
4
5)1
9
Exercise 7The yield of a tree plantation is given by f(x)=
-20 + 20 x + 28 x2
39 x5, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)3
4
2)5
7
3)29
12
4)29
4
5)11
9
Exercise 8
Study the differentiability of the function f(x)=ⅇx+1 - 3 cos(x + 1) - 3 x ≤ -14 x - 3 (x + 2) log(x + 2) - 1 -1 < x < 1-2 ⅇx-1 + 5 - 9 log(3) 1 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-1.
4) The function is differentiable for all points except for x=1.
5) The function is differentiable for all points except for x=-1 and x=1.
78
Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 78028660
Exercise 1Study the shape properties of the f(x)=5 + 24 x2 + 16 x3 + 3 x4
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To find the maximun and minimum points of the function,
try (with Ruffini) the points -2, -1, 0, 1, 2. To solve this exercise
it is necessary to determine the increasing and decreasing intervals.
Exercise 2Determine the derivative of the function f(t)=
t + 2 (t + cos(t)) and compute its value at the point t=0.
1) f'(0)=2 2) f'(0)=3 3) f'(0)=-2 4) f'(0)=-1
Exercise 3
Compute the limit: limx→1
3
2- 2 x +
x2
2+ Log[x]
-1 + 3 x - 3 x2 + x3
1) -1
2) -∞
3) ∞
4) 0
5)1
3
6) 1
7) -2
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Exercise 4
Compute the limit: limx→-1-2 - 3 x + x3
3 + 7 x + 5 x2 + x3
1) -2
2) -∞
3) 0
4) ∞
5) -1
6) 1
7) -3
2
Exercise 5Between the months t=0 and t=7
, the funds in certain account (in millions of euros) are given by the function F(t)=
6 - 6 t2 + 2 t3.
Determine the interval where the temperature oscillates between the months t=4 and t=7.
1) It oscillates between 33 and 394.
2) It oscillates between 38 and 398.
3) It oscillates between -2 and 6.
4) It oscillates between 45 and 401.
5) It oscillates between -2 and 398.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of18 x
8 + 49 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)17
4
2)29
19
3)1
2
4)4
49
5)9
20
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Exercise 7The yield of a tree plantation is given by f(x)=
-3 + 47 x + 30 x2
45 x2, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)21
17
2)5
4
3)1
2
4)6
47
5)8
9
Exercise 8Study the differentiability of the function f(x)=
-sin(x + 3) - 2 cos(x + 3) + 2 x ≤ -3-3 x + 2 sin(x + 3) + 2 cos(x + 3) - 11 -3 < x < -2
sin(x + 2) + cos(x + 2) + 2 -3 + sin(1) + cos(1) -2 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=-3.
4) The function is differentiable for all points except for x=-2.
5) The function is differentiable for all points except for x=-3 and x=-2.
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Mathematics 1 - ADE/FyCo - 2019/2020List of exercises 02-Derivatives for identity number: 300530374
Exercise 1Study the shape properties of f(x)=5 + 20 x3 - 15 x4 + 2 x6
to decide which amongst the following ones is the representation of the function.
1) 2)
3) 4)
Big point: maximum Small point: minimum
Red line: convexity Green line: concavity
Indication: To solve this exercise it is necessary to determine the concavity and
convexity interval. To find the inflexion points separating the cancavity and
convexity intervals, check (by means of Ruffini) with the points -2, -1, 0, 1, 2.
Exercise 2Determine the derivative of the function f(t)=
3 ⅇⅇsin(t)
+ cossin(t) and compute its value at the point t=0.
1) f'(0)=0 2) f'(0)=-3 3) f'(0)=3 ⅇ 4) f'(0)=2
Exercise 3
Compute the limit: limx→0-1 + ⅇx
3
- x3
x6
1) -2
2) ∞
3) -1
4)1
2
5) -∞
6) 1
7) 0
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Exercise 4
Compute the limit: limx→-2-8 - 4 x + 6 x2 + 5 x3 + x4
-12 - 8 x + x2 + x3
1) -∞
2) -2
3
3) -2
4) 0
5) ∞
6) -1
7) 1
Exercise 5Between the months t=2 and t=9
, the true value of the shares of a company (in euros) are given by the function C(t)=
620 + 378 t - 48 t2 + 2 t3.
Determine the interval where the value oscillates between the months t=2 and t=8.
1) It oscillates between 1209 and 1600.
2) It oscillates between 1200 and 1600.
3) It oscillates between 1209 and 1601.
4) It oscillates between 1190 and 1595.
5) It oscillates between 1592 and 1600.
Exercise 6In a fish farm, data of precedings years reveal that, if the initial number of fishes is x
(x in thousands), after one month they reproduce until a number of49 x
16 + 9 xfishes.
What number of fishes leads to the largest increase in the number of animals, f(x)-x?
1)4
3
2)13
4
3)2
15
4)29
9
5)27
17
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Exercise 7The yield of a tree plantation is given by f(x)=
-48 + 39 x + 11 x2
31 x9, where x is the distance in meters between two trees.
What is the distance x that leads to the largest yield.
1)7
6
2)12
11
3) 13
4)3
5
5)40
3
Exercise 8Study the differentiability of the function f(x)=
ⅇx-3 - 2 sin(3) sin(x) - 2 cos(3) cos(x) + 5 x ≤ 31
6x2 + 15 3 < x < 6
2 ⅇx-6 +13
26 ≤ x
1) The function is differentiable for all points.
2) The function is not differentiable at any point.
3)The function is differentiable for all points except for x=3.
4) The function is differentiable for all points except for x=6.
5) The function is differentiable for all points except for x=3 and x=6.
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