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Introduction to Calculus Preparation for MAT 1475: Single Variable Calculus I
Supported by
NSF MSP Grant # 1102729
NSF STEP Grant #0622493 and
City Tech Black Male Initiative
January 2014
New York City College of Technology
Dr. Janet Liou-Mark
Dr. Urmi Ghosh-Dastidar Department of Mathematics
Christopher Chan, Yiming Yu, and Karmen Yu
dddddddd
Travion Joseph, Karmen Yu, and Yiming Yu
Table on Contents
I. Finding Limits Graphically and Numerically 1
II. One-Sided Limits Graphically and Numerically 8
III. Evaluating Limits Analytically 17
IV. The Derivative of a Function and Tangent Lines 21
V. Differentiation Rules 32
VI. The Derivative of Trigonometric Functions 42
VII. Implicit Differentiation 47
VIII. Solutions 51
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 1
Section 1: Finding Limits Graphically and Numerically 1. A function )(xf has a limit if the value of )(xf is the same as you approach x from both sides.
Lxfax
=β
)(lim If )(xf equals L as x approaches β a β from the left and right directions.
Example: Find1
1lim
3
1 β
β
β x
x
x, 1x
x .75 .9 .99 .999 1 1.001 1.01 1.1 1.25
)(xf 2.313 2.170 2.970 2.997 Error 3.003 3.030 3.310 3.813
As x approaches 1 from the left and right directions, the values of )(xf approach 3.
Therefore the limit is 3 or 31
1lim
3
1=
β
β
β x
x
x
x Approaches 1 from the right
x Approaches 1 from the left
)(xf Approaches 3 )(xf Approaches 3
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 2
2. Create a table with values on both sides of the value x approaches and draw a graph to confirm the limit for each question.
a) 152
5lim
25 ββ
β
β xx
x
x
x 4.75 4.9 4.99 4.999 5 5.001 5.01 5.1 5.25
)(xf
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 3
c) x
x
x
sinlim
0β
x 0
)(xf
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 4
d) 4
1lim
4 +ββ xx
x - 4
)(xf
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 5
e) x
x
x
33lim
0
β+
β
x 0
)(xf
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 6
f) 22 )2(
1lim
ββ xx
x 2
)(xf
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 7
3) Limits That Fail to Exist a) Behavior that differs from the right and left
b) Unbounded behavior
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 8
c) Oscillating behavior
Section 2: One-Sided Limits, Infinite Limits, and Continuity 1) One-Sided Limits
Left-hand limit: )(lim xfax ββ
means: compute the limit of )(xf as x approaches a from the left
Right-hand limit: )(lim xfax +β
means: compute the limit of )(xf as x approaches a from the right
Example: The left hand limit: 3)(lim =ββ
xfax
The right hand limit: 1)(lim =+β
xfax
2) The Lxfax
=β
)(lim if and only if Lxfax
=ββ
)(lim and Lxfax
=+β
)(lim .
Explain: _________________________________________________________ _________________________________________________________
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 9
3) Infinite Limits
Infinite Limits: =β
)(lim xfax
Infinite Limits: β=β
)(lim xfax
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 10
7) Draw a sketch of each of the following infinite limits.
=ββ
)(lim xfax
=+β
)(lim xfax
β=ββ
)(lim xfax
β=+β
)(lim xfax
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 11
8) Find the limits and values.
π(2)
limπ₯β2+
π(π₯)
limπ₯β2β
π(π₯)
limπ₯β2
π(π₯)
π(β1)
limπ₯ββ1+
π(π₯)
limπ₯ββ1β
π(π₯)
limπ₯ββ1
π(π₯)
π(β3)
limπ₯ββ3+
π(π₯)
limπ₯ββ3β
π(π₯)
limπ₯ββ3
π(π₯)
π(1)
limπ₯β1+
π(π₯)
limπ₯β1β
π(π₯)
limπ₯β1
π(π₯)
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 12
9) Find the limits and values.
π(2)
limπ₯β2+
π(π₯)
limπ₯β2β
π(π₯)
limπ₯β2
π(π₯)
π(β1)
limπ₯ββ1+
π(π₯)
limπ₯ββ1β
π(π₯)
limπ₯ββ1
π(π₯)
π(β3)
limπ₯ββ3+
π(π₯)
limπ₯ββ3β
π(π₯)
limπ₯ββ3
π(π₯)
π(1)
limπ₯β1+
π(π₯)
limπ₯β1β
π(π₯)
limπ₯β1
π(π₯)
MAT 1475 INTRODUCTION TO CALCULUS I
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10) Find the limits and values.
π(7)
limπ₯β7+
π(π₯)
limπ₯β7β
π(π₯)
limπ₯β7
π(π₯)
π(5)
lim
π₯β5+π(π₯)
limπ₯β5β
π(π₯)
limπ₯β5
π(π₯)
π(1)
limπ₯β1+
π(π₯)
limπ₯β1β
π(π₯)
limπ₯β1
π(π₯)
π(β4)
lim
π₯ββ4+π(π₯)
limπ₯ββ4β
π(π₯)
limπ₯ββ4
π(π₯)
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 14
11) Find the limits and values.
π(β7)
limπ₯ββ7+
π(π₯)
limπ₯ββ7β
π(π₯)
limπ₯ββ7
π(π₯)
π(β6)
limπ₯ββ6+
π(π₯)
limπ₯ββ6β
π(π₯)
limπ₯ββ6
π(π₯)
π(β3)
limπ₯ββ3+
π(π₯)
limπ₯ββ3β
π(π₯)
limπ₯ββ3
π(π₯)
π(3)
limπ₯β3+
π(π₯)
limπ₯β3β
π(π₯)
limπ₯β3
π(π₯)
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 15
12) Sketch the graph of a function )(xf that
at 3β=a shows that )3(βf is not defined,
at 2=a shows that )(lim2
xfxβ
does not exist,
at 7=a shows that )7()(lim7
fxfx
β
, but is continuous elsewhere
13) Sketch the graph of a function )(xf that
at 6β=a shows that )(lim6
xfx ββ
does not exist,
at 0=a shows that )0(f is not defined,
at 5=a shows that )5()(lim5
fxfx
β
, but is continuous elsewhere
MAT 1475 INTRODUCTION TO CALCULUS I
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14) Continuity at a Point A function f is continuous at c if the following three conditions are met.
1. )(cf is defined.
2. )(lim xfcxβ
exists.
3. )()(lim cfxfcx
=β
15) Discontinuity
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 17
Section 3: Evaluating Limits Analytically
1) Properties of Limits
Suppose c is a constant and the limits )(lim xfaxβ
and )(lim xgaxβ
exists.
Limit Properties Example:
Let 22)( xxf = and xxg =)(
1. Sum )(lim)(lim)]()([lim xgxfxgxfaxaxax βββ
+=+
2. Difference )(lim)(lim)]()([lim xgxfxgxfaxaxax βββ
β=β
3. Scalar multiple
)(lim)]([lim xfcxcfaxax ββ
=
4. Product )(lim)(lim)]()([lim xgxfxgxfaxaxax βββ
β’=
5. Quotient
)(lim
)(lim
)(
)(lim
xg
xf
xg
xf
ax
ax
ax
β
β
β= if 0)(lim
βxg
ax
6. Power n
ax
n
axxfxf )](lim[)]([lim
ββ=
where n is a positive integer
7. ccax
=β
lim
8. axax
=β
lim
9. nn
axax =
βlim where n is a positive integer
10. nn
axax =
βlim where n is a positive integer
If n is even, we assume that a > 0
11. Root nax
n
axxfxf )(lim)(lim
ββ=
where n is a positive integer
If n is even, we assume that 0)(lim β
xfax
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 18
2) Strategies for Finding Limits Method 1: The Direct Substitution Property is valid for all polynomial and rational functions with nonzero denominators. Example: Find the limit.
a) 22
3)54(lim +
βx
x b)
2
76lim
2
1 +
β+
ββ x
xx
x
Method 2: Dividing Out Technique β Factor and divide out any common factors Example: Find the limit.
a) 5
103lim
2
5 +
β+
ββ x
xx
x b)
x
xx
x
7lim
2
0
+
β
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 19
Method 3: Rationalizing Technique β Rationalizing the numerator of a fractional expression Example: Find the limit.
a) x
x
x
11lim
0
β+
β b)
x
x
x
33lim
0
β+
β
3) Find the limit.
a) )5(lim 23
4xx
x+β
ββ
b) 42lim6
+β
xx
c) xx
x
x 3
5lim
21 β
β
ββ
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 20
d) 49
7lim
27 β
β
ββ x
x
x
e) 158
103lim
2
2
5 +β
ββ
β xx
xx
x
f) 2
8lim
3
2 β
β
β x
x
x
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 21
g) x
xx
5
1
5
1
lim0
++
β
h) x
x
x
66lim
0
β+
β
i) 4
35lim
4 β
β+
β x
x
x
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 22
Section 4: The Derivative of a Function
Calculus is primarily the study of change. The basic focus on calculus is divided in to two
categories, Differential Calculus and Integral Calculus. In this section we will introduce differential
calculus, the study of rate at which something changes.
Consider the example at which π₯ is to be the independent variable and π¦ the dependent variable.
If there is any change βπ₯ in the value of π₯, this will result in a change βπ¦ in the value of π¦. The
resulting change in π¦ for each unit of change in π₯ remains constant and is called the slope of the
line.
The slope of a straight line is represented as:
βπ¦
βπ₯=
π¦2βπ¦1
π₯2βπ₯1=
Change in the π¦ coordinate
Change in the π₯ coordinate
TANGENT LINES
Calculus is concerned with the rate of change that is not constant. Therefore, it is not possible to
determine a slope that satisfies every point of the curve. The question that calculus presents is:
βWhat is the rate of change at the point P?β And we can find the slope of the tangent line to the
curve at point P by the method of differentiation. A tangent line at a given point to a plane curve is
a straight line that touches the curve at that point.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 23
SECANT LINES
Like a tangent line, a secant line is also a straight line; however a secant line passes through two
points of a given curve.
Therefore we must consider an infinite sequence of shorter intervals of βπ₯, resulting in an infinite
sequence of slopes. We define the tangent to be the limit of the infinite sequence of slopes. The
value of this limit is called the derivative of the given function.
The slope of the tangent at P = limβπ₯β0
βπ¦
βπ₯
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 24
THE DEFINITION OF THE DERIVATIVE
THE DIFFERENCE QUOTIENT
To find the slope of the tangent line to the function π¦ = π(π₯) at, we must choose a point of
tangency, (π₯, π(π₯)) and a second point (π₯ + β, π(π₯ + β)), where β = βπ₯.
The derivative of a function f at x is defined as:
πβ²(π₯) = limβπ₯β0
βπ¦
βπ₯= lim
βπ₯β0
π(π₯+βπ₯)βπ(π₯)
βπ₯ = lim
ββ0
π(π₯+β)βπ(π₯)
β
Consider the example where the function π(π₯) = 7π₯ + 11. Find πβ²(π₯) by using the definition of
the derivative.
π(π₯) = 7π₯ + 11
π(π₯ + β) = 7(π₯ + β) + 11
Now by the definition of the derivative:
πβ²(π₯) = limββ0
7(π₯ + β) + 11 β (7π₯ + 11)
β
πβ²(π₯) = limββ0
7π₯ + 7β + 11 β 7π₯ β 11
β
πβ²(π₯) = limββ0
7β
β
πβ²(π₯) = limββ0
(7)
πβ²(π₯) = 7
Therefore the derivative of π(π₯) is 7.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 25
Consider the example where the function π(π₯) = 3π₯2 + 6π₯ β 9. Find πβ²(π₯) by using the
definition of the derivative.
π(π₯) = 3π₯2 + 6π₯ β 9
π(π₯ + β) = 3(π₯ + β)2 + 6(π₯ + β) β 9
Now by the definition of the derivative:
πβ²(π₯) = limββ0
3(π₯ + β)2 + 6(π₯ + β) β 9 β (3π₯2 + 6π₯ β 9)
β
πβ²(π₯) = limββ0
3(π₯2 + 2π₯β + β2) + 6π₯ + 6β β 9 β 3π₯2 β 6π₯ + 9
β
πβ²(π₯) = limββ0
3π₯2 + 6π₯β + 3β2 + 6π₯ + 6β β 9 β 3π₯2 β 6π₯ + 9
β
πβ²(π₯) = limββ0
6π₯β + 3β2 + 6β
β
πβ²(π₯) = limββ0
(6π₯ + 3β + 6)
πβ²(π₯) = 6π₯ + 3(0) + 6
πβ²(π₯) = 6π₯ + 6
Therefore the derivative of π(π₯) is 6π₯ + 6 .
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 26
3) Find the derivative of the following functions using the definition of derivative.
a) xxxf 43)( 2 β=
b) xxg 51)( β=
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 27
c) 1
6)(
+=
y
yyh
d) xxxp +=)(
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 28
e) 2
1)(
ttk =
4) What does derivative of )(xf mean graphically?
Answer: It is the slope ( m ) of the tangent line of the graph )(xfy = at the point ),( yx
Tangent line
y = f(x)
),( yx
The slope of the tangent line is given by
tan0 0
( ) ( ) ( ) ( )lim lim
( )gent
h h
f x h f x f x h f xm
x h x hβ β
+ β + β= =
+ β = )(' xf if this limit exists.
This is actually the derivative of )(xf .
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 29
5) Find the slope of the tangent line to the curve 2)( 2 += xxf at the point )3,1(β using the definition
of the derivative, and find the equation of the tangent line.
By definition, the slope of the tangent line at any point is given by )(' xf .
Therefore )(' xf equals to the following:
0
2 2
0
2 2 2
0
2
0
0
0
( ) ( )lim
(( ) 2) ( 2)lim
( 2 2) 2lim
2lim
(2 )lim
lim( 2 )
2 0
2
h
h
h
h
h
h
f x h f xm
h
x h xm
h
x xh h xm
h
xh hm
h
h x hm
h
m x h
m x
m x
β
β
β
β
β
β
+ β=
+ + β +=
+ + + β β=
+=
+=
= +
= +
=
Now this is the slope of the tangent at a point ))(,( xfx of the graph. Since the line is tangent at )3,1(β ,
we have to evaluate m at )3,1(β . Therefore, 2)1(2 β=β=m .
The slope of the tangent at )3,1(β is 2β .
To find the equation of the tangent line to the curve 2)( 2 += xxf use the point-slope formula to find
the equation:
))1((2)3( βββ=β xy
223 ββ=β xy
12 +β= xy
The equation of the tangent is 12 +β= xy .
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 30
6) Find the slope of the tangent line to the curve xxxf +β= 23)( at the point )10,2( β using the
definition of the derivative, and find the equation of the tangent line.
7) Find the slope of the tangent line to the curve 3)( xxf = at the point )8,2( ββ using the definition of
the derivative, and find the equation of the tangent line.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 31
8) Find the slope of the tangent line to the curve 6
1)(
β=
xxf at the point )1,5( β using the definition
of the derivative, and find the equation of the tangent line.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 32
Section 5: Differentiation Rules
1) The Differentiation Formulas
1.
Derivative of a Constant Function 0)( =c
dx
d
2.
The Power Rule
1)( β= nn nxxdx
d
where n is any real number
3.
The Constant Multiple Rule )()]([ xf
dx
dcxcf
dx
d=
4.
The Sum Rule )()()]()([ xg
dx
dxf
dx
dxgxf
dx
d+=+
5.
The Difference Rule )()()]()([ xg
dx
dxf
dx
dxgxf
dx
dβ=β
6.
The Product Rule )]([)()]([)()]()([ xf
dx
dxgxg
dx
dxfxgxf
dx
d+=
or in prime notation
fggffg +=)(
7.
The Quotient Rule
2)]([
)]([)()]([)(
)(
)(
xg
xgdx
dxfxf
dx
dxg
xg
xf
dx
dβ
=
or in prime notation
2g
gffg
g
f β=
8.
The Chain Rule )())](([))](([ xg
dx
dxgf
dx
dxgf
dx
d=
2) Examples:
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 33
1. Derivative of a Constant Function 0)( =cdx
d
a) 60)( =xf b) π¦ =8800718
369+
40π
47β 62768.32
2. The Power Rule 1)( β= nn nxx
dx
d where n is any real number
c) 5xy = d)
63)( β= xxg
3. The Constant Multiple Rule )()]([ xfdx
dcxcf
dx
d=
e) xy 4β= f) 73)( xxp =
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 34
4.
The Sum Rule )()()]()([ xg
dx
dxf
dx
dxgxf
dx
d+=+
g) 3
2
2
1
42 xxy += h) 142)( 5 ++= β xxxt
5. The Difference Rule )()()]()([ xgdx
dxf
dx
dxgxf
dx
d+=+
i) xxxy 752 24 βββ= β j)
17 34)( ββ β= xxxh
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 35
6. The Product Rule
)]([)()]([)()]()([ xfdx
dxgxg
dx
dxfxgxf
dx
d+=
or in prime notation fggffg +=)(
k) )32(4 += xxy l) )6)(73()( 2 xxxxa +β=
7. The Quotient Rule
2)]([
)]([)()]([)(
)(
)(
xg
xgdx
dxfxf
dx
dxg
xg
xf
dx
dβ
=
or in prime notation 2g
gffg
g
f β=
m) 13 +
=x
xy n)
xx
xxq
23
9)(
2
2
β=
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 36
8. The Chain Rule )())](([))](([ xgdx
dxgf
dx
dxgf
dx
d=
o) 82 )3( += xy p)
1000)5005(2)( β= xxs
3) Find the derivative of the following functions:
a) 235)( 63 β+= ββ xxxf
b) 36 xxy β=
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 37
c) )7)(34()( β+= ttth
d) 9
5)(
2 β
+=
x
xxp
e) 2
2 5)(
ttts +=
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 38
f) 2
23 84
x
xxy
+β=
g) 6
1
2
3
3)( xxxm +=
β
h) 432 )142(4 ββ= xxy
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 39
i) x
xxxf
125)(
2 +β=
j) xx
y 92
3+=
k) 1003 )42()( xxxv ββ=
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 40
4)
a) Find the slope of the tangent line to the curve xxxf +β= 23)( at the point )10,2( β using the
differentiation formulas, and find the equation of the tangent line.
b) Find the slope of the tangent line to the curve 3)( xxf = at the point )8,2( ββ using the
differentiation formulas, and find the equation of the tangent line.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 41
c) Find the slope of the tangent line to the curve 6
1)(
β=
xxf at the point )1,5( β using the
differentiation formulas, and find the equation of the tangent line.
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 42
Section 6: The Derivative of Trigonometric Functions
Derivative of all six Trigonometric Functions
1. Sine π
ππ₯sin(π₯) = cos (π₯)
2. Cosine π
ππ₯cos(π₯) = βsin (π₯)
3. Tangent π
ππ₯tan(π₯) = sec2(π₯)
4. Cosecant π
ππ₯cot(π₯) = β csc2(π₯)
5. Secant π
ππ₯sec(π₯) = sec(x) tan (π₯)
6. Cotangent π
ππ₯csc(π₯) = β csc(x) cot (π₯)
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 43
Example: Find the derivative of the trigonometric function π(π₯) = 2π₯3cos (π₯) using the rules of differentiation.
A derivative that requires the Product Rule
π
ππ₯(2π₯3 cos(π₯)) = [
π
ππ₯(2π₯3)] cos(π₯) + 2π₯3 [
π
ππ₯cos(π₯)]
Therefore, the derivative of the function π(π₯) is 6π₯2 cos(π₯) β 2π₯3sin (π₯). 2) Find the derivative of the following trigonometric functions using the rules of differentiation.
a) π¦ = 7 sin(π₯) β π₯
b) π(π₯) = π₯cos(2π₯2)
MAT 1475 INTRODUCTION TO CALCULUS I
NEW YORK CITY COLLEGE OF TECHNOLOGY 44
c) π(π₯) = tan ( 3x
4 )
d) π¦ =cos(x)
2sin (β3π₯)
e) π(π₯) = 5cos3(π₯) β sin(2π₯)
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f) π¦ = βsin2(π₯) + 5
g) π£(π₯) = tan(βπ₯3 + 2)
h) π(π₯) = 2π₯2sec2(8π₯)
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i) π(π₯) = 3π₯2 sin(π₯) sec(5π₯)
j) Given π(π₯) = tan(π₯) and π(π₯) = sec2(π₯). Show π
ππ₯π(π₯) = π(π₯).
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Section 7: Implicit Differentiation
IMPLICIT DIFFERENTIATON
Consider the two equations:
π¦ = π₯2 β 5 and π₯2 + π¦2 = 4
The first equation defines π¦ as a function of π₯ explicitly. For each value of π₯, the equation returns
an explicit formula π¦ = π(π₯) for finding the corresponding value of π¦. However, the second
equation does not define a function, as it fails the vertical line test.
If we look at a general case, consider the example:
π₯2 + π¦2 = π2
This is an equation of a circle with the radius π. In order for us to solve for the derivative ππ¦
ππ₯, we
may treat π¦ as a function of π₯ and apply the chain rule. This process is called implicit
differentiation.
π₯2 + π¦2 = π2
ππ¦
ππ₯π₯2 +
ππ¦
ππ₯π¦2 =
ππ¦
ππ₯π2
2π₯ + 2π¦ππ¦
ππ₯= 0
ππ¦
ππ₯= β
π₯
π¦
The derivative of an implicit function π(π¦) is defined as
π
ππ₯π(π¦) = πβ²(π¦)π¦β²(π₯)
Example: Find π¦β²(π₯) for π₯2π¦2 β 2π₯ = 4 β 4π¦ using the method of implicit differentiation.
π
ππ₯(π₯2π¦2 β 2π₯) =
π
ππ₯(4 β 4π¦)
2π₯π¦2 + π₯2(2π¦)π¦β²(π₯) β 2 = β4π¦β²(π₯)
(2π₯π¦2 + 4)π¦β²(π₯) = 2 β 2π₯π¦2
π¦β²(π₯) =(2 β 2π₯π¦2)
2π₯2π¦ + 4
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1) Find the derivative π¦β²(π₯) using the method of implicit differentiation for the following.
a) π₯5 + π¦4 = 9
b) 5βπ¦ + β2π₯ = 4
c) π₯π¦ β 3π¦ = π₯
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d) 2π₯3π¦ + 6π¦ = 1 β 2π₯
e) βπ¦π₯ β 3π¦3 = 20
f) sin(π¦) β π¦2 = 15
MAT 1475 INTRODUCTION TO CALCULUS I
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g) β2π₯ + π¦ β 5π₯2 = π¦
h) tan(5π¦) β π₯π¦2 = 3π₯
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Section 8: Solutions Section 1
2.a. 4.75 0.129032
2.b. -0.1 0.998334
4.9 0.126582 -0.01 0.999983 4.99 0.125156 -0.001 1 4.999 0.125016 -0.0001 1 5 undefined 0 undefined 5.001 0.124984 0.0001 1 5.01 0.124844 0.001 1 5.1 0.123457 0.01 0.999983 5.25 0.121212 0.1 0.998334 Hole @ (5,
1
8), VA@ π₯ = β3, HA @ π¦ = 0
Hole @ (0,1)
2.c. -4.1 -10
2.d. -0.1 0.291122
-4.01 -100 -0.01 0.288916 -4.001 -1000 -0.001 0.288699 -4.0001 -10000 -0.0001 0.288678 -4 undefined 0 undefined -3.9999 10000 0.0001 0.288673 -3.999 1000 0.001 0.288651 -3.99 100 0.01 0.288435 -3.9 10 0.1 0.286309
VA @ π₯ = β4, HA @ π¦ = 0
Hole @ (0,β3
6), HA @ π¦ = 0, Endpoint @ (β3,
β3
3)
2.e. 1.9 100
1.99 10000 1.999 1000000 1.9999 1E+08 2 undefined 2.0001 1E+08 2.001 1000000 2.01 10000 2.1 100 VA @ π₯ = 2, HA @ π¦ = 0
7. Answers may vary.
MAT 1475 INTRODUCTION TO CALCULUS I
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8. π(2) = 4
limπ₯β2+
π(π₯) = 4
limπ₯β2β
π(π₯) = 4
limπ₯β2
π(π₯) = 4
π(β1) = 2
limπ₯ββ1+
π(π₯) = 1
limπ₯ββ1β
π(π₯) = 2
limπ₯ββ1
π(π₯) = π·ππΈ
π(β3) = 3
limπ₯ββ3+
π(π₯) = β2
limπ₯ββ3β
π(π₯) = β2
limπ₯ββ3
π(π₯) = β2
π(1) = undefined
limπ₯β1+
π(π₯) = 1
limπ₯β1β
π(π₯) = 1
limπ₯β1
π(π₯) = 1
9. π(2) = β1
limπ₯β2+
π(π₯) = 2
limπ₯β2β
π(π₯) = β2
limπ₯β2
π(π₯) = π·ππΈ
π(β1) = 1
limπ₯ββ1+
π(π₯) = 1
limπ₯ββ1β
π(π₯) = 1
limπ₯ββ1
π(π₯) = 1
π(β3) = 1
limπ₯ββ3+
π(π₯) = 3
limπ₯ββ3β
π(π₯) = 1
limπ₯ββ3
π(π₯) = π·ππΈ
π(1) = undefined
limπ₯β1+
π(π₯) = β1
limπ₯β1β
π(π₯) = β1
limπ₯β1
π(π₯) = β1
10. π(7) = 7
limπ₯β7+
π(π₯) = β1
limπ₯β7β
π(π₯) = β1
limπ₯β7
π(π₯) = β1
π(5) = 3
lim
π₯β5+π(π₯) = β5
limπ₯β5β
π(π₯) = 3
limπ₯β5
π(π₯) = π·ππΈ
π(1) = 3
limπ₯β1+
π(π₯) = 3
limπ₯β1β
π(π₯) = 6
limπ₯β1
π(π₯) = π·ππΈ
π(β4) = 3
limπ₯ββ4+
π(π₯) = 6
limπ₯ββ4β
π(π₯) = 3
limπ₯ββ4
π(π₯) = π·ππΈ
11. π(β7) = β1
limπ₯ββ7+
π(π₯) = β1
limπ₯ββ7β
π(π₯) = β1
limπ₯ββ7
π(π₯) = β1
π(β6) = undefined
limπ₯ββ6+
π(π₯) = 3
limπ₯ββ6β
π(π₯) = ββ
limπ₯ββ6
π(π₯) = π·ππΈ
π(β3) = 3
limπ₯ββ3+
π(π₯) = 3
limπ₯ββ3β
π(π₯) = 3
limπ₯ββ3
π(π₯) = 3
π(3) = 6
limπ₯β3+
π(π₯) = 6
limπ₯β3β
π(π₯) = β3
limπ₯β3
π(π₯) = β3
12. Answers may vary. 13. Answers may vary. Section 2 2. Answers may vary. Sample response: βA limit exists if and only if the left-hand and
right-hand limit exists.β 7. Answers may vary. 12. Answers may vary. 13. Answers may vary.
MAT 1475 INTRODUCTION TO CALCULUS I
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Section 3 (*Answers may vary depending on constant) 1.1. lim
π₯βπ(2π₯2 + π₯) = lim
π₯βπ2π₯2 + lim
π₯βππ₯ 1.2. lim
π₯βπ(2π₯2 β π₯) = lim
π₯βπ2π₯2 β lim
π₯βππ₯
1.3 * limπ₯βπ
(12.86(2π₯2)) = 12.86 limπ₯βπ
(2π₯2) 1.4 limπ₯βπ
(2π₯2 β π₯) = limπ₯βπ
2π₯2 β limπ₯βπ
π₯
1.5 limπ₯βπ
(2π₯2
π₯) =
limπ₯βπ
2π₯2
limπ₯βπ
π₯ 1.6 lim
π₯βπ(2π₯2)5 = (lim
π₯βπ(2π₯2))
5
1.7. * limπ₯βπ
π = π 1.8. * limπ₯βπ
π₯ = π 1.9. * limπ₯β6
π₯3 = 63 = 216
1.10. * limπ₯β32
βπ₯5
= β325
= 2 1.11. * limπ₯βπ
β2π₯25= βlim
π₯βπ2π₯25
2.1.a. 1681 2.1.b. 0 2.2.a. β7 2.2.b. 7 2.3.a. 1
2 2.3.b.
β3
6
3.a. 144 3.b. 4 3.c. β3
2 3.d. DNE 3.e.
7
2 3.f. 12
3.g. DNE 3.h. β6
12
Section 4
3.a. 6π₯ β 4 3.b. β5
2β1β5π₯ 3.c.
6
(π¦+1)2 3.d. 1
2βπ₯+ 1
3.e. β2
π‘3 6. derivative: πβ²(π₯) = β6π₯ + 1 slope: β11 tangent: π¦ = β11π₯ + 12 7. derivative: πβ²(π₯) = β3π₯2 slope: 12 tangent: π¦ = 12π₯ + 16 8. derivative: πβ²(π₯) = β(π₯ β 6)β2 slope: β1 tangent: π¦ = βπ₯ + 4 Section 5 (Note: Equivalent solutions may be calculated, depending on rules applied.) 2.1.a. 0 2.1.b. 0 2.2.c. 5π₯4 2.2.d. β18π₯β7 2.3.e. β4 2.3.f. 21π₯6
2.4.g. π₯β1
2 +8
3π₯β
1
3 2.4.h. 4 β 10π₯β6 2.5.i. 8π₯β5 β 10π₯ β 7 2.5.j. 3π₯β2 β 28π₯β8
2.6.k. 10π₯4 + 12π₯3 2.6.l. 9π₯2 + 22π₯ β 42 2.7.m. 1
(3π₯+1)2 2.7.n. β18
(3π₯β2)2
2.8.o. 16π₯(π₯2 + 3)7 2.8.p. 10000(5π₯ β 500)999
3.a. β15π₯β4 β 18π₯β7 3.b. 3π₯β1
2 β1
3π₯β
2
3 3.c. 8π‘ β 25 3.d. βπ₯2+10π₯+9
(π₯2β9)2
3.e. 2π‘ β 10π‘β3 3.f. 1 β 16π₯β3 3.g. 1
2π₯β
5
6 β3
2π₯β
5
2
3.h. β896π₯(π₯3 β 7)3(π₯3 β 1) 3.i. 5 β π₯β2 3.j. 9 β2
3π₯β
4
3
3.k. 2100π₯2(206π₯ β 3)(1 β 2π₯)99 4.a. derivative: πβ²(π₯) = β6π₯ + 1 slope: β11 tangent: π¦ = β11π₯ + 12 4.b. derivative: πβ²(π₯) = β3π₯2 slope: 12 tangent: π¦ = 12π₯ + 16 4.c. derivative: πβ²(π₯) = β(π₯ β 6)β2 slope: β1 tangent: π¦ = βπ₯ + 4
MAT 1475 INTRODUCTION TO CALCULUS I
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Section 6 2.a. π¦β² = 7 cos(π₯) β 1 2.b. πβ²(π₯) = cos(2π₯2) β 4π₯2sin (2π₯2)
2.c. πβ²(π₯) =3
4sec2 (
3π₯
4) 2.d. π¦β² =
sin(π₯) sin(3π₯)+3 cos(3π₯) cos(π₯)
2 sin2(3π₯)
2.e. πβ²(π₯) = 15cos2(π₯) sin(π₯) β 2 cos(2π₯) =15
2cos(π₯) sin(2π₯) β 2cos (2π₯)
2.f. π¦β² = sin(π₯) cos(π₯)
βsin2(π₯)+5=
sin(2π₯)
2βsin2(π₯)+5 2.g. π£β²(π₯) =
3π₯2 sec2(βπ₯3+2)
2βπ₯3+2
2.h. πβ²(π₯) = 4π₯ sec2(8π₯) + 32π₯2 sec2(8π₯) tan(8π₯) 2.i. πβ²(π₯) = 6π₯ sin(π₯) sec(5π₯) + 3π₯2 cos(π₯) sec(5π₯) + 15π₯2 sin(π₯) sec (5π₯) tan(5π₯) 2.j. Answers may vary.
Suggested ideas include quotient rule using tan(π₯) =sin(π₯)
cos(π₯) or use limit definition of
πβ(π₯) using angle sum identities.
Section 7
1.a. β5π₯4
4π¦3 1.b. ββ2π₯π¦
5π₯ 1.c.
1βπ¦
π₯β3 1.d. β
3π₯2π¦+1
π₯3+3
1.e. (18π¦3
βπ₯π¦+π₯π¦)
324π₯π¦5βπ₯2 1.f. 0, y β {y β β: cos(y) β 2y} 1.g. 20π₯β2π₯+π¦β2
1β2β2π₯+π¦
1.h. π¦2+3
5 sec2(5π¦)β2π₯π¦