MA16100I Fall 2015 Final Review Problems

The following a a collection of problems to help prepare for the final exam. There are a lot of problems here,so don’t feel like you have to do them all. Also, these problems are not meant to be a representation of thefinal exam problems themselves. Final exam problems may be worded or phrased differently than what isbelow. For an accurate set of problems that the final exam problems would look like, check out the PastExam Archive on the Purdue Math Department webpage.

(I) Solve the following trigonometric equations.

(1) tanx = 1

(2) sinx = −1

2

(3) cosx =

√3

2

(4) cscx =√

2

(5) cotx = −√

3

(6) secx = 1

(7) cscx = 0

(8) tan2 x− tanx = 0

(9) sinx = cosx

(10) 2 cos2 x− cosx = 1

(11) sin 2x = sinx

(12) cosx− cos 2x = 1

(13) sin2 x+ cosx = 1

(14) sinx cosx =1

2

(II) Sketch a graph of each of the following trigonometric functions. What is the period of each function?

(1) y = sin 2x

(2) y = −3 cosπx

(3) y = tan(x

2

)(4) y = 2 csc 3x

(5) y = cotπx

3(6) y = − sec 4x

(7) y = 2 sin(

2x− π

2

)− 2

(8) y = cos(

3x− π

2

)+ 1

(III) Simplify the following expressions involving inverse trigonometric functions.

(1) tan−1(−1)

(2) sin−1(

1√2

)(3) cos−1

(√3

2

)(4) csc−1 2

(5) csc(sec−1 2

)+ cos

(tan−1

(−√

3))

(6) sec(cot−1

√3 + csc−1(−1)

)(7) sec−1

(sec(−π

6

))(8) sec

(tan−1

x

2

)(9) cos

(sin−1 x

)(10) sin

(sec−1

√x2 + 4

x

)

(IV) Simplify.

(1) 253/2

(2) 8−4/3

(3)

(4

9

)1/2

(4) (−27)2/3

(5)[642/3

] [64−3/2

](6) 813/4 − 2432/5

(7)(−3x−2

) (4x4)

(8)

(2x3) (

3x2)

(x2)3

(9) 2x(3x2)2

(10)(x2yz3

) (−2xz2

) (x3y−2

)(11)

(1

3x4y−3

)−2(12)

(−2xy2

)5( x7

8y3

)

1

(V) Expand out each of the logarithmic expressions.

(1) ln√x(x+ 1)

(2) ln

√1

t(t+ 1)

(3) ln

(θ + 5

θeθ

)(4) log2

(x√x2 + 1

(x+ 1)2/3

)

(5) ln

√(x+ 1)10

(2x+ 1)5

(6) ln(ex

2√2x+ 1

)(7) ln 3

√x(x+ 2)

x2 + 1

(8) ln

(ex

2√x(x+ 1)

)

(VI) Graph each of the following exponential/logarithmic functions.

(1) f(x) = 3x

(2) f(x) = 5−x

(3) f(x) = 2 (6x)

(4) f(x) = 1− 2x

(5) f(x) = 42−x − 3

(6) f(x) = −5 + ex−4

(7) f(x) = log10 x

(8) f(x) = log2 (2x)

(9) f(x) = log5 (x− 4) + 3

(10) f(x) = 2 log3 (x+ 1)

(11) f(x) = ln(x2)

(12) f(x) = ln (x− 1)

(VII) Solve the following exponential/logarithmic equations.

(1) 32x−1 = 27

(2) 32x = 4(23x)

(3) 324 =1

2x32x

(4)

(1

8

)x−1= 23−2x

2

(5) 42x−x2

=1

64

(6) 24x =23x

2

3x2

576(7) 2x = 3

(8) 43x−2 = 19

(9) 10 = 7 + 6e2x

(10) log7 (1− x) = 3

(11) 2x = e2

(12)16ex

ex + 4= 10

(13)4

1 + 3e−2x= 1

(VIII) Given the following point pairs, find the values of C and a such that the graph of y = Cax containsthe two points.

(1) (0, 2), (1, 12) (2) (1,−12), (−1,−3/4) (3) (2, 1/2), (−1, 4) (4) (0,−5), (−1,−15)

2

(IX) Given the graph, find the one-sided and two-sided limits at the given points. At which of these pointsare the graphs continuous?

(1) c = −1, 0, 1, 2, 3

(2) c = −1, 0, 1, 2, 3

(3) c = −1, 0, 1, 2, 3, 4, 5, 6, 7

(4) c = −1, 0, 1, 2, 3, 4, 5 (its difficult to see, butthere is a solid point at (3, 0))

(X) Given the limits of f(x) and g(x), find the specified limits.

(1) Suppose limx→−2

f(x) = −7 and limx→−2

g(x) = 0. Find

(a) limx→−2

3f(x)

(b) limx→−2

f(x)g(x)

(c) limx→−2

cos (g(x))

(d) limx→−2

(f(x))2

(e) limx→−2

f(x)

g(x)− 7

(f) limx→−2

|f(x)|

(2) Suppose limx→0

f(x) = 1/2 and limx→0

g(x) =√

2. Find

(a) limx→0−g(x)

(b) limx→0

f(x) + g(x)

(c) limx→0

x+ f(x)

(d) limx→0

g(x)f(x)

(e) limx→0

1

f(x)

(f) limx→0

f(x) cosx

x− 1

(3) Suppose limx→1

f(x) = 1 and limx→1

g(x) = 8. Find

(a) limx→1

f(x)− g(x)

(b) limx→1

f(x)− x2(c) lim

x→1

g(x) sinx

x

(d) limx→1

ln (f(x))

(e) limx→1

f(x) + g(x)

f(x)− 2g(x)

(f) limx→1

g (f(x))

3

(XI) Find the following limits.

(1) limx→−7

(2x+ 5)

(2) limx→12

(10− 3x)

(3) limx→2

(−x2 + 5x− 2

)(4) lim

x→5

4

x− 7

(5) limx→0

(2z − 8)1/3

(6) limx→−5

x2 + 3x− 10

x+ 5

(7) limx→2

x3 − 8

x4 − 16

(8) limx2 − 4x+ 4

x3 + 5x2 − 14x

(a) x→ 0

(b) x→ 1

(c) x→ 2

(9) limx2 + x

x5 + 2x4 + x3

(a) x→ −1

(b) x→ 0

(c) x→ 1

(10) limx→1

1−√x

1− x

(11) limx→−1

√x2 + 8− 3

x+ 1

(12) limx→1

4x− x2

2−√x

(13) limx→2

sin

(1

x− 1

2

)(14) lim

x→0sec[cosx+ π tan

( π

4 secx

)− 1]

(15) limx→0

sin(√

2x)

√2x

(16) limx→0

sin 3y

4y

(17) limx→0

x

sin 3x

(18) limx→0

tan 2x

x

(19) limx→0

x csc 2x

cos 5x

(20) limx→0

(1 + 3x)1/x

(21) limx→0

ln (1 + x)2/x

(22) limx→0

(1 + 3x)1/(5x)

(23) lim1

x2 − 4

(a) x→ 2+

(b) x→ 2−

(c) x→ −2+

(d) x→ −2−

(24) limx2 − 3x+ 2

x3 − 2x2

(a) x→ 2+

(b) x→ 2−

(c) x→ 2

(d) x→ 0+

(e) x→ 0

(25) limx2 − 3x+ 2

x3 − 4x

(a) x→ 2+

(b) x→ −2+

(c) x→ 0−

(d) x→ 1+

(e) x→ 0

4

(XII) Where are the following functions continuous? For each point of discontinuity, determine the type(i.e. removable, infinite, step, infinite oscillation).

(1) f(x) =

1 x ≤ −1−x −1 < x < 01 x = 0−x 0 < x < 11 x ≥ 1

.

(2) f(x) =

0 x ≤ −11/x 0 < |x| < 10 x = 11 x > 1

.

(3) f(x) =

−x x ≤ 0

sin(πx

)0 < x < 2

(x− 2)2 2 ≤ x < 4√8x− x2 4 ≤ x ≤ 8

.

(4) f(x) =

0 x ≤ 11/(x− 1) 1 < x < 2x− 2 2 ≤ x < 3(x− 2)2 3 ≤ x

.

(5) f(x) =

0 x = 0

sin

(1

x

)x 6= 0

.

(XIII) Find the vertical asymptotes of the following functions.

(1) f(x) =5

2x

(2) f(x) =1

x2 − 4

(3) f(x) =ex

x+ 1

(4) f(x) =x− 1

x2 + 4x− 5

(5) f(x) =cosx

x− 1

(6) f(x) =sinx

x(7) f(x) = tanx

(8) f(x) = cotx

(9) f(x) = secx

(10) f(x) = cscx

(11) f(x) = lnx

(12) f(x) = ln(x2 + 1)

(13) f(x) = ln(x2 − 1)

(14) f(x) =x2 − 3x+ 2

x3 − 4x

(15) f(x) = 2− 1

x1/3

(16) f(x) =−2

(x− 3)2

(XIV) Find the derivative using the definition of the derivative. (I know, I know, using the definition is apain, and we will be spending the next few weeks talking about the derivative rules. But it is goodpractice with algebra to evaluate via the definition.)

(1) f(x) = 4− x2

(2) f(x) = (x− 1)2 + 1

(3) f(x) =1

x2

(4) f(x) =1− x

2x

(5) f(x) =√

3x

(6) f(x) = x− 1

x

(7) f(x) =1√x+ 1

(8) f(x) = x+√x

(9) f(x) =1

(x− 1)2

(XV) Find the equation for the tangent line of the given function at the given point.

(1) f(x) = x2 + 1 at c = 2

(2) f(x) = x− 2x2 at c = 1

(3) f(x) =x

x− 2at c = 3

(4) f(x) =8

x2at c = 2

(5) f(x) = x3 at c = 2

(6) f(x) = x3 + 3x at c = 1

(7) f(x) =√x+ 1 at c = 3

(8) f(x) =√

5x− 1 at c = 2

(9) f(x) =x− 1

x+ 1at c = 0

5

(XVI) Find the first and second derivatives, dy/dx and d2y/dx2.

(1) f(x) = 4− x2

(2) f(x) = (x− 1)2 + 1

(3) f(x) =1

x2

(4) f(x) =1− x

2x

(5) f(x) =√

3x

(6) f(x) = x− 1

x

(7) f(x) =1√x+ 1

(8) f(x) = x+√x

(9) f(x) =1

(x− 1)2

(10) y =x2 − 1

x2 + 1

(11) y = (1− x)(1 + x2)−1

(12) y =5x+ 1

2√x

(13) y =x3 + 7

x

(14) y =(x− 1)(x2 + x+ 1)

x3

(15) y =

(1 + 3x

3x

)(3− x)

(16) y = (x+ 1)(x− 1)(x2 + 1)

(17) y =x2 + 3

(x− 1)3 + (x+ 1)3

(18) y = tanx

(19) y = cscx

(20) y = x2 − secx+ 1

(21) y = 4− x2 sinx

(22) y = x sinx+ cosx

(23) y = secx cscx

(24) y = 6x2 cotx csc(2x)

(25) y = (2x+ 1)5

(26) y =(

1− x

7

)−7(27) y =

(x

5+

1

5x

)5

(28) y = sec(tanx)

(29) y = 5(cosx)−4

(30) y = x2 sin4 x+ x cos−2 x

(31) y = (5− 2x)−3 +1

8

(2

x+ 1

)4

(32) y = x tan (2√x) + 7

(33) y = sec (√x) tan

(1x

)(34) y = cos

(5 sin

(x3

))(35) y = 4 sin

(√x+√x)

(36) x3 + y3 = 18xy

(37) y2 =x− 1

x+ 1

(38) x+ tan(xy) = 0

(39) y2 cos(

1y

)= 2x+ 2y

(40) y = x7 + x√

7− 1π+1

(41) y = 2 tan2 x− sec2 x

(42) xy + 2x+ 3y = 1

(43)√xy = 1

(44) y = x (sin (lnx) + cos (lnx))

(45) y = ln

√(x+ 1)5

(x+ 2)20

(46) y =

√x

x+ 1

(47) y =x sinx√

secx

(48) y =x√x2 + 1

(x+ 1)2/3

(49) y = 3

√x(x+ 1)(x− 2)

(x2 + 1)(2x+ 3)

(50) ln y = ey sinx

(51) ln(xy) = ex+y

(52) e2x = sin(x+ 3y)

(53) tan y = ex + ln y

(54) y = esin x(ln(x2)

+ 1)

(55) y =e1/x

x2

(56) y = (2x+ 5)−1/2

(57) y = x(x2 + 1)−1/3

(58) y = cos−1(x2)

(59) y = sec−1 5x

(60) y = tan−1 (lnx)

6

(61) y = ln(x2 + 4

)− x tan−1

(x2

)(62) y = 1

2 sinh (2x+ 1)

(63) y = sechx (1− ln sechx)

(64) y = 2√x tanh

√x

(65) y = 2x

(66) y = (x+ 1)x

(67) y = (lnx)ln x

(68) y = (√x)x

(XVII) Find the equation for the tangent line of the given graph at the given point.

(1) y = x2 − 4 at c = 2

(2) y = sinx at c = π/3

(3) y = 1 + cosx at c = −π/3

(4) y = x+9

xat c = −3

(5) y = (x+ 1)3 at c = −2

(6) x2 + y2 = 25 at (3,−4)

(7) 6x2 + 3xy + 2y2 + 17y − 6 = 0 at (−1, 0)

(8) x2 cos2 y − sin y = 0 at (0, π)

(9) x2y2 = 9 at (−1, 3)

(10) y4 = y2 − x2 at

(√3

4,

1

2

)(11) y2(2− x) = x3 at (1, 1)

(12) y4 − 4y2 = x4 − 9x2 at (3, 2)

(13) x+√xy = 6 at (4, 1)

(14) x3/2 + 2y3/2 = 17 at (4, 1)

(XVIII) Find points of the graph that have a horizontal tangent line.

(1) y = −x3 + 3x2 − 3x

(2) y = (x4/4)− x3 + x2(3) y = x+ sinx

(4) y = x− cotx

(5) x2 + xy + y2 = 7

(6) x3 + y3 − 9xy = 0

(XIX) Answer the following word problems.

(1) A rock is thrown vertically at a rate of 64 ft/s from the edge of a 196 ft cliff. Suppose the heightof the rock is given by

s(t) = −16t2 + v0t+ s0.

(a) What is the velocity and acceleration of the rock after t seconds?

(b) What is the velocity and acceleration of the rock at t = 1? t = 5?

(c) How high does the rock get?

(d) What is the speed of the rock when it hits the bottom of the cliff?

(2) Now assume that the rock encounters some air resistance while traveling, and its height is nowgiven by

s(t) = ke−t − 16t+ a.

(a) What is the velocity and acceleration of the rock after t seconds?

(b) What is the velocity and acceleration of the rock at t = 1? t = 5?

(c) How high does the rock get?

(3) The power P of an electric circuit is related to the circuit’s resistance R and current i by theequation P = Ri2.

(a) Write an equation to determine dP/dt if R and i are functions of t.

(b) For a given circuit, at a specific moment in time, the power and resistance of a resistor is 1 kWand 16 kΩ, respectively. If the power is increasing at 100 W/s and the current is increasing at0.1 A/s, then how fast is R changing?

7

(4) Water is flowing at the rate of 6 m3/min from a reservoir shaped like a hemispherical bowl ofradius 13 m. The volume of water remaining is given by

V =1

3πy2(3R− y)

where R is the radius of the hemisphere and y is the height of the water level from the base.

(a) At what rate is the water level changing when the water is 8 m deep?

(b) What is the radius r of the water’s surface when the water is ym deep?

(c) At what rate is the radius r changing when the water is 8 m deep?

(5) A spherical iron ball 8 in in diameter is coated with a layer of ice of uniform thickness. If the icemelts at the rate of 10 in3/min, how fast is the thickness of the ice decreasing when it is 2 in thick?How fast is the surface area changing? (For a sphere, the volume is V = (4/3)πr3, and the surfacearea is S = 4πr2).

(6) One morning, the shadow of an 80 ft building on level ground is 60 ft long. At the moment inquestion, the angle θ the sun makes with the ground is increasing at the rate of 0.27/min. Atwhat rate is the shadow decreasing? (Careful about units!)

(7) A particle moves along the parabola x = y2 in the first quadrant in such a way that its y-coordinate(measured in meters) increases at a steady 10 m/sec. How fast is the distance the particle is fromthe point (1/4, 0) changing when x = 3? How fast is the angle between the +x-axis and the linejoining the particle to the point (1/4, 0) changing when x = 3?

(8) A man 6 ft tall walks at the rate of 5 ft/sec toward a streetlight that is 16 ft above the ground. Atwhat rate is the tip of his shadow moving when he is 10 ft from the base of the light?

(9) A light shines form the top of a pole 50 ft high. A ball is dropped from the same height form apoint 30 ft away from the light. How fast is the shadow of the ball moving along the ground 1/2sec later? Assume the ball falls a distance s = 16t2 ft in t seconds.

(10) Two airplaces are flying along straight line course that intersect at right angles. Plane A isapproaching the intersection point at a speed of 442 mph. Plane B is flying away from theintersection at 481 mph. At what rate is the distance between the planes changing when A is 5miles from the intersection point and B is 12 miles from the intersection point?

(11) At what rate is the distance between the tip of the second hand and the 12 o’clock mark (the topof the circular clock) changing when the second hand points to 4 o’clock?

(12) Two ships are steaming straight away from a point O along routes that make a 120 angle. ShipA moves at 14 mph, and ship B moves at 21 mph. How fast are the ships moving apart from eachother when OA = 5 and OB = 3 miles?

(13) Water is flowing at a rate of 50 m3/min from a shallow concrete conical reservoir (vertex down) ofbase radius 45 m and height 6 m. How fast is the water level falling when the water is 5 m deep?How fast is the area of the surface of the water changing then?

(XX) Find the absolute extrema of each function on the given interval.

(1) f(x) = 4− x2 on [−3, 1]

(2) f(x) = − 1

x2on [1, 2]

(3) f(x) = x+1

xon

[1

2, 3

](4) f(x) =

x2

x+ 1on

[−1

2, 1

]

(5) f(x) =√

4− x2 on [−1, 2]

(6) f(x) = secx on[−π

3,π

6

](7) f(x) = x2ex on [−3, 1]

(8) f(x) = x ln(x+ 3) on [0, 3]

8

(XXI) Find the intervals of increasing and decreasing, concavity, inflection points, and relative extrema foreach function. Make a sketch of the graph of the function.

(1) f(x) = x4 − 6x3 − 5

(2) f(x) = x4

(3) f(x) = (x+ 1)(x− 1)2

(4) f(x) =x+ 1

(x− 1)2

(5) f(x) = x5 − 5x4

(6) f(x) = x1/5

(7) f(x) = (2− x2)3/2

(8) f(x) =x3

3x+ 1

(9) f(x) =x2 − 3

x− 2

(10) f(x) = x√

8− x2

(11) f(x) = x2 lnx

(XXII) For each of the following graphs, find the intervals of concavity and inflection points if

(a) the graph is of f (b) the graph is of f ′

(1) (2)

(3)

(XXIII) Find the following limits.

(1) limx→∞

x2 − 2x+ 5

2x2 + 5x+ 1=

1

2

(2) limx→−∞

2x+ 1

3x2 + 2x− 7= 0

(3) limx→∞

5− 2x3/2

3x2 − 4

(4) limx→−∞

x√x2 − x

(5) limx→−∞

√x4 − 1

x3 − 1

(6) limx→−∞

2x

(x6 − 1)1/3

(7) limx→∞

cosx

(8) limx→∞

sin1

x

(9) limx→∞

sin 3x

x

(10) limx→∞

x sin1

x

(11) limx→−∞

2− x+ sinx

x+ cosx

(12) limx→∞

[8

x− arctanx

](13) lim

x→∞

1

ex + 1

(14) limx→−∞

1

ex + 1

(15) limx→±∞

[x+

√x2 + 3

](16) lim

x→±∞

[3x−

√9x2 − x

](17) lim

x→∞[ln(x+ 1)− lnx]

9

(18) limx→0

sin 5x

x

(19) limx→0

x sinx

1− cosx

(20) limx→0

x2

ln (secx)

(21) limx→0

x2x

2x − 1

(22) limx→0

sinx2

x

(23) limx→3

x− 3

x2 − 3

(24) limx→∞

√9x+ 1√x+ 1

(25) limx→∞

1

x lnx

∫ x

1

ln t dt

(26) limx→1+

x1/(1−x)

(27) limx→∞

x1/ ln x

(28) limx→0+

xx

(29) limx→0

2sin x − 1

ex − 1

(30) limx→0+

e−1/x lnx

(31) limx→1+

√x2 − 1

sec−1 x

10

(XXIV) Find the indefinite integrals.

(1)

∫4x5 dx

(2)

∫ex dx

(3)

∫4 cosx dx

(4)

∫−5

xdx

(5)

∫sec2 x dx

(6)

∫− secx tanx dx

(7)

∫3 sinx dx

(8)

∫ (3x2 − 2x+ 1

)dx

(9)

∫cos (7θ + 5) dθ

(10)

∫x2 sin

(x3)dx

(11)

∫1

cos2(2θ)dθ

(12)

∫sin4 t cos t dt

(13)

∫3z dz

3√z2 + 1

(14)

∫ (x4 − 2x2 + 8x− 2

) (x3 − x+ 2

)dx

(15)

∫18 tan2 x sec2 x(

2 + tan3 x)2 dx

(16)

∫sin 3x dx

(17)

∫x3(x4 − 1

)2dx

(18)

∫ √3− 2s ds

(19)

∫4y dy√2y2 + 1

(20)

∫tan7 x

2sec2

x

2dx

(21)

∫r4(

7− r5

10

)3

dr

(22)

∫ (e3x + 5e−x

)dx

(23)

∫e−√r

√rdr

(24)

∫esecπt secπt tanπt dt

(25)

∫er

1 + erdr

(26)

∫dx

1 + ex

(27)

∫2x

x2 − 5dx

(28)

∫x2 + 2x+ 3

x3 + 3x2 + 9xdx

(29)

∫dx

2√x+ 2x

(30)

∫dx√

−x2 + 4x− 1

11

(XXV) Evaluate the definite integral.

(1)

∫ 3

0

√y + 1 dy

(2)

∫ 1

−1r√

1− r2 dr

(3)

∫ π/4

0

tanx sec2 x dx

(4)

∫ π

0

3 cos2 x sinx dx

(5)

∫ 1

0

5r

(4 + r2)2 dr

(6)

∫ 4

1

10√v(

1 + v3/2)2 dv

(7)

∫ π/6

0

cos−3 2θ sin 2θ dθ

(8)

∫ 1

0

(4y − y2 + 4y3 + 1

)−2/3 (12y2 − 2y + 4

)dy

(9)

∫ π/2

π/4

(1 + ecot θ

)csc2 θ dθ

(10)

∫ √lnπ

0

2xex2

cos(ex

2)dx

(11)

∫ −2−3

dx

x

(12)

∫ π/2

−π/2

4 cos θ

3 + 2 sin θdθ

(13)

∫ 2

1

2 lnx

xdx

(14)

∫ 4

2

dx

x (lnx)2

(15)

∫ 2

0

x2 − 2

x+ 1dx

(16)

∫ 1

0

x− 1

x+ 1dx

(XXVI) Solve the differential equations.

(1)dy

dx= 12x

(3x2 − 1

)3, y(1) = 3

(2)dy

dt= et sin

(et − 2

), y(ln 2) = 0

(XXVII) Solve the following problems.

(1) A hard-boiled egg at 98 C is put in a sink of 18C water. After 5 minutes, the egg’s temperatureis 38C. Assuming that the water has not warmed appreciably, how much longer will it take theegg to reach 20C?

(2) An aluminum beam was brought from the outside cold into a machine shop where the temperaturewas held at 65 F. After 10 minutes, the beam warmed to 35 F and after another 10 minutes itwas 50 F.. Estimate the beam’s initial temperature.

(XXVIII) Find the derivative of the following functions.

(1) f(x) =

∫ √x0

cos t dt

(2) f(x) =

∫ x4

0

√u du

(3) f(x) =

∫ sin x

x

3t2 dt

(4) f(x) =

∫ x

0

√1 + t2 dt

(5) f(x) =

∫ x2

tan x

cos√t dt

(6) f(x) =

∫ tan x

− tan x

dt

1 + t2

12