Question 1

Find the y-intercept of the tangent line to the curve y = x $3$ at the point (1, 1).

Accepted Answer

A)  - 4 
B) 4 
C)  - 16 
D) 2 
E)  - 8
F)8
G) - 2
H)16 
A)  - 4 
B) 4 
C)  - 16 
D) 2 
E)  - 8
F)8
G) - 2
H)16

Question 2

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated surface area.

Accepted Answer

The answer of The diameter of a sphere is measured...

Question 3

Find $\frac { d y } { d x } .$ (a) $y = 7 ^ { 3 } + x ^ { 3 }$ (b) $y = x ^ { 50 } + 25 x - 1$ (c) $y = \left( x ^ { 3 } - 3 \sqrt [ 3 ] { x } \right) / x$ (d) $y = 2 e ^ { x } + x ^ { e }$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 4

According to the theory of relativity, the mass of an object at speed v is given by $m ( v ) = \frac { m _ { 0 } } { \sqrt { 1 - \frac { v ^ { 2 } } { c ^ { 2 } } } }$ where c is the speed of light and $m _ { 0 }$ is the mass of the object when it is at rest. Find $m ^ { \prime } ( v )$ .

Accepted Answer

The answer of According to the theory of relativity, the...

Question 5

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated volume.

Accepted Answer

The answer of The diameter of a sphere is measured...

Question 6

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

Accepted Answer

The answer of Find the x-coordinate(s) of the point(s) where...

Question 7

If $x y = 8$ , find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point ( $- 2$ , 4)?

Accepted Answer

A)  $- 4$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)4
H)8 
A)  $- 4$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)4
H)8

Question 8

Find the exact value of $\tan ^ { - 1 } ( \cos \pi )$ .

Accepted Answer

A)  $- \frac { \pi } { 2 }$ 
B)  $- \frac { \pi } { 4 }$ 
C) 0 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { 3 \pi } { 4 }$
F) $\frac { 5 \pi } { 6 }$
G) $\frac { \pi } { 2 }$
H) $\pi$ 
A)  $- \frac { \pi } { 2 }$ 
B)  $- \frac { \pi } { 4 }$ 
C) 0 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { 3 \pi } { 4 }$
F) $\frac { 5 \pi } { 6 }$
G) $\frac { \pi } { 2 }$
H) $\pi$

Question 9

Given $f ( 3 ) = 5 , f ^ { \prime } ( 3 ) = 1.1 , g ( 3 ) = - 4 \text { and } g ^ { \prime } ( 3 ) = 0.7 \text {, find the value of } ( f / g ) ^ { \prime } ( 3 ) \text {. }$

Accepted Answer

A) 0.025 
B) 0.49375 
C)  $- 0.49375$ 
D)  $- 0.025$ 
E) 1.975
F) $- 1.975$
G)0.5625
H) $- 0.5625$ 
A) 0.025 
B) 0.49375 
C)  $- 0.49375$ 
D)  $- 0.025$ 
E) 1.975
F) $- 1.975$
G)0.5625
H) $- 0.5625$

Question 10

Consider the two functions $f ( x ) = e ^ { \sin x }$ and $g ( x ) = \sin \left( e ^ { x } \right)$ (a) Which, if either, of these functions is periodic? Justify your answer.(b) For each function, consider the limit as x increases without bound. Does either function also increase without bound like an exponential function? Explain.(c) Where, if anywhere, does each function have an x-intercept? Justify your answer.(d) Where, if anywhere, does each function have a horizontal tangent line? Justify your answer.(e) Where, if anywhere, does each function attain its maximum value? its minimum value? Justify your answers.

Accepted Answer

(a) Since @#IMG-DLM& is periodic.
(b) No, @#IMG-DLM& and @#IMG-DLM& Bo

Question 11

If $f ( x ) = \sqrt [ 5 ] { x } + x ^ { - 18 }$ , find $f ^ { \prime } ( 1 )$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 5 }$ 
C)  $- \frac { 8 } { 5 }$ 
D)  $- \frac { 2 } { 5 }$ 
E)  $- \frac { 1 } { 5 }$ 
F) $\frac { 4 } { 5 }$

G) $\frac { 8 } { 5 }$

H) $- \frac { 4 } { 5 }$ 
A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 5 }$ 
C)  $- \frac { 8 } { 5 }$ 
D)  $- \frac { 2 } { 5 }$ 
E)  $- \frac { 1 } { 5 }$ 
F) $\frac { 4 } { 5 }$

G) $\frac { 8 } { 5 }$

H) $- \frac { 4 } { 5 }$

Question 12

Differentiate the following functions:
(a) $f ( x ) = \frac { 3 } { x ^ { 2 } } + x ^ { 2 }$ (b) $f ( t ) = \frac { 4 \pi } { 3 } t ^ { 3 }$ (c) $f ( s ) = e ^ { x }$ (d) $f ( r ) = ( 4 r - 3 ) ^ { 2 }$

Accepted Answer

The answer of Differentiate the following functions:
(a) \(f ( x...

Question 13

Use differentials to approximate $\sqrt { 16.2 }$ .

Accepted Answer

A) 4.026 
B) 4.03 
C) 4.025 
D) 4.05 
E) 4.015
F)0.02498
G)4.0185
H)4.0245 
A) 4.026 
B) 4.03 
C) 4.025 
D) 4.05 
E) 4.015
F)0.02498
G)4.0185
H)4.0245

Question 14

Find an equation of the tangent line to the curve $x ^ { 2 } y ^ { 4 } + x y = 2$ at (1, 1).

Accepted Answer

The answer of Find an equation of the tangent line...

Question 15

Given $f ( x ) = x ^ { 4 } e ^ { x }$ , find the value(s) of x where $f ^ { \prime \prime } ( x ) = 0$ .

Accepted Answer

The answer of Given \(f ( x ) = x...

Question 16

f and g are functions whose graphs are given below. Let $u ( x ) = f ( x ) \cdot g ( x ) \text { and } v ( x ) = \frac { f ( x ) } { g ( x ) }$   (a) Find $u ^ { \prime } ( 1 )$ .(b) Find $u ^ { \prime } ( - 1 )$ .(c) Find $v ^ { \prime } ( - 1 )$

Accepted Answer

The answer of f and g are functions whose graphs...

Question 17

If $f ( x ) = \frac { x } { x - 1 } , \text { find a formula for } f ^ { ( n ) } ( x )$

Accepted Answer

A)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
B)  $f ^ { ( n ) } = n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
C)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - n }$ 
D)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - ( n - 1 ) }$ 
E)  $f ^ { ( n ) } = ( - 1 ) ^ { n + 1 } n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
F)
$f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { n + 1 }$

G) $f ^ { ( n ) } = n ! ( x - 1 ) ^ { - n + 1 }$

H) $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x + 1 ) ^ { - ( n + 1 ) }$ 
A)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
B)  $f ^ { ( n ) } = n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
C)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - n }$ 
D)  $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { - ( n - 1 ) }$ 
E)  $f ^ { ( n ) } = ( - 1 ) ^ { n + 1 } n ! ( x - 1 ) ^ { - ( n + 1 ) }$ 
F)
$f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x - 1 ) ^ { n + 1 }$

G) $f ^ { ( n ) } = n ! ( x - 1 ) ^ { - n + 1 }$

H) $f ^ { ( n ) } = ( - 1 ) ^ { n } n ! ( x + 1 ) ^ { - ( n + 1 ) }$

Question 18

Let $f ( x ) = \ln ( \ln ( x ) )$ . Find the value of $f ^ { \prime } ( e )$ .

Accepted Answer

A)  $\frac { 1 } { e ^ { 2 } }$ 
B)  $e ^ { 2 }$ 
C) 1 + e 
D)  $\frac { 1 } { e }$ 
E) ln 2 
F)1
G)e
H)0 
A)  $\frac { 1 } { e ^ { 2 } }$ 
B)  $e ^ { 2 }$ 
C) 1 + e 
D)  $\frac { 1 } { e }$ 
E) ln 2 
F)1
G)e
H)0

Question 19

There are two tangent lines to the curve $y = 4 x - x ^ { 2 }$ that pass through (2, 5). Find an equation for each of the lines.

Accepted Answer

@#IMG-DLM& . We seek points (x

Question 20

Find $\frac { d y } { d x }$ .(a) $y = \ln \sqrt { \frac { 4 x - 7 } { x ^ { 2 } + 2 x } }$ (b) $y = ( \ln ( \sec x ) ) ^ { 4 }$ (c) $y = \left( 1 + \frac { 1 } { x } \right) ^ { 2 x }$

Accepted Answer

(a) @#IMG-DLM& = @#IMG-DLM& (b

Find the y-intercept of the tangent line to the curve y = x $3$ at the point (1, 1).

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated surface area.

Find $\frac { d y } { d x } .$ (a) $y = 7 ^ { 3 } + x ^ { 3 }$ (b) $y = x ^ { 50 } + 25 x - 1$ (c) $y = \left( x ^ { 3 } - 3 \sqrt [ 3 ] { x } \right) / x$ (d) $y = 2 e ^ { x } + x ^ { e }$

According to the theory of relativity, the mass of an object at speed v is given by $m ( v ) = \frac { m _ { 0 } } { \sqrt { 1 - \frac { v ^ { 2 } } { c ^ { 2 } } } }$ where c is the speed of light and $m _ { 0 }$ is the mass of the object when it is at rest. Find $m ^ { \prime } ( v )$ .

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated volume.

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

If $x y = 8$ , find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point ( $- 2$ , 4)?

Find the exact value of $\tan ^ { - 1 } ( \cos \pi )$ .

Given $f ( 3 ) = 5 , f ^ { \prime } ( 3 ) = 1.1 , g ( 3 ) = - 4 \text { and } g ^ { \prime } ( 3 ) = 0.7 \text {, find the value of } ( f / g ) ^ { \prime } ( 3 ) \text {. }$

If $f ( x ) = \sqrt [ 5 ] { x } + x ^ { - 18 }$ , find $f ^ { \prime } ( 1 )$

Differentiate the following functions: (a) $f ( x ) = \frac { 3 } { x ^ { 2 } } + x ^ { 2 }$ (b) $f ( t ) = \frac { 4 \pi } { 3 } t ^ { 3 }$ (c) $f ( s ) = e ^ { x }$ (d) $f ( r ) = ( 4 r - 3 ) ^ { 2 }$

Use differentials to approximate $\sqrt { 16.2 }$ .

Find an equation of the tangent line to the curve $x ^ { 2 } y ^ { 4 } + x y = 2$ at (1, 1).

Given $f ( x ) = x ^ { 4 } e ^ { x }$ , find the value(s) of x where $f ^ { \prime \prime } ( x ) = 0$ .

If $f ( x ) = \frac { x } { x - 1 } , \text { find a formula for } f ^ { ( n ) } ( x )$

Let $f ( x ) = \ln ( \ln ( x ) )$ . Find the value of $f ^ { \prime } ( e )$ .

There are two tangent lines to the curve $y = 4 x - x ^ { 2 }$ that pass through (2, 5). Find an equation for each of the lines.

Find $\frac { d y } { d x }$ .(a) $y = \ln \sqrt { \frac { 4 x - 7 } { x ^ { 2 } + 2 x } }$ (b) $y = ( \ln ( \sec x ) ) ^ { 4 }$ (c) $y = \left( 1 + \frac { 1 } { x } \right) ^ { 2 x }$

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Exam 3: Differentiation Rules

Find the y-intercept of the tangent line to the curve y = x 333 at the point (1, 1).

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated surface area.

Find dydx.\frac { d y } { d x } .dxdy​. (a) y=73+x3y = 7 ^ { 3 } + x ^ { 3 }y=73+x3 (b) y=x50+25x−1y = x ^ { 50 } + 25 x - 1y=x50+25x−1 (c) y=(x3−3x3)/xy = \left( x ^ { 3 } - 3 \sqrt [ 3 ] { x } \right) / xy=(x3−33x​)/x (d) y=2ex+xey = 2 e ^ { x } + x ^ { e }y=2ex+xe

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated volume.

Find the x-coordinate(s) of the point(s) where the tangent to the curve y=x+cos⁡x,0≤x≤2πy = x + \cos x , 0 \leq x \leq 2 \piy=x+cosx,0≤x≤2π has zero slope.

If xy=8x y = 8xy=8 , find the value of d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ at the point ( −2- 2−2 , 4)?

Find the exact value of tan⁡−1(cos⁡π)\tan ^ { - 1 } ( \cos \pi )tan−1(cosπ) .

If f(x)=x5+x−18f ( x ) = \sqrt [ 5 ] { x } + x ^ { - 18 }f(x)=5x​+x−18 , find f′(1)f ^ { \prime } ( 1 )f′(1)

Differentiate the following functions: (a) f(x)=3x2+x2f ( x ) = \frac { 3 } { x ^ { 2 } } + x ^ { 2 }f(x)=x23​+x2 (b) f(t)=4π3t3f ( t ) = \frac { 4 \pi } { 3 } t ^ { 3 }f(t)=34π​t3 (c) f(s)=exf ( s ) = e ^ { x }f(s)=ex (d) f(r)=(4r−3)2f ( r ) = ( 4 r - 3 ) ^ { 2 }f(r)=(4r−3)2

Use differentials to approximate 16.2\sqrt { 16.2 }16.2​ .

Find an equation of the tangent line to the curve x2y4+xy=2x ^ { 2 } y ^ { 4 } + x y = 2x2y4+xy=2 at (1, 1).

Given f(x)=x4exf ( x ) = x ^ { 4 } e ^ { x }f(x)=x4ex , find the value(s) of x where f′′(x)=0f ^ { \prime \prime } ( x ) = 0f′′(x)=0 .

If f(x)=xx−1, find a formula for f(n)(x)f ( x ) = \frac { x } { x - 1 } , \text { find a formula for } f ^ { ( n ) } ( x )f(x)=x−1x​, find a formula for f(n)(x)

Let f(x)=ln⁡(ln⁡(x))f ( x ) = \ln ( \ln ( x ) )f(x)=ln(ln(x)) . Find the value of f′(e)f ^ { \prime } ( e )f′(e) .

There are two tangent lines to the curve y=4x−x2y = 4 x - x ^ { 2 }y=4x−x2 that pass through (2, 5). Find an equation for each of the lines.

Find dydx\frac { d y } { d x }dxdy​ .(a) y=ln⁡4x−7x2+2xy = \ln \sqrt { \frac { 4 x - 7 } { x ^ { 2 } + 2 x } }y=lnx2+2x4x−7​​ (b) y=(ln⁡(sec⁡x))4y = ( \ln ( \sec x ) ) ^ { 4 }y=(ln(secx))4 (c) y=(1+1x)2xy = \left( 1 + \frac { 1 } { x } \right) ^ { 2 x }y=(1+x1​)2x

Functions and Models

Limits and Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the y-intercept of the tangent line to the curve y = x $3$ at the point (1, 1).

Find $\frac { d y } { d x } .$ (a) $y = 7 ^ { 3 } + x ^ { 3 }$ (b) $y = x ^ { 50 } + 25 x - 1$ (c) $y = \left( x ^ { 3 } - 3 \sqrt [ 3 ] { x } \right) / x$ (d) $y = 2 e ^ { x } + x ^ { e }$

Find the x-coordinate(s) of the point(s) where the tangent to the curve $y = x + \cos x , 0 \leq x \leq 2 \pi$ has zero slope.

If $x y = 8$ , find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point ( $- 2$ , 4)?

Find the exact value of $\tan ^ { - 1 } ( \cos \pi )$ .

If $f ( x ) = \sqrt [ 5 ] { x } + x ^ { - 18 }$ , find $f ^ { \prime } ( 1 )$

Differentiate the following functions: (a) $f ( x ) = \frac { 3 } { x ^ { 2 } } + x ^ { 2 }$ (b) $f ( t ) = \frac { 4 \pi } { 3 } t ^ { 3 }$ (c) $f ( s ) = e ^ { x }$ (d) $f ( r ) = ( 4 r - 3 ) ^ { 2 }$

Use differentials to approximate $\sqrt { 16.2 }$ .

Find an equation of the tangent line to the curve $x ^ { 2 } y ^ { 4 } + x y = 2$ at (1, 1).

Given $f ( x ) = x ^ { 4 } e ^ { x }$ , find the value(s) of x where $f ^ { \prime \prime } ( x ) = 0$ .

If $f ( x ) = \frac { x } { x - 1 } , \text { find a formula for } f ^ { ( n ) } ( x )$

Let $f ( x ) = \ln ( \ln ( x ) )$ . Find the value of $f ^ { \prime } ( e )$ .

There are two tangent lines to the curve $y = 4 x - x ^ { 2 }$ that pass through (2, 5). Find an equation for each of the lines.

Find $\frac { d y } { d x }$ .(a) $y = \ln \sqrt { \frac { 4 x - 7 } { x ^ { 2 } + 2 x } }$ (b) $y = ( \ln ( \sec x ) ) ^ { 4 }$ (c) $y = \left( 1 + \frac { 1 } { x } \right) ^ { 2 x }$