Question 1

Evaluate the derivative of the function at the given point. $f ( t ) = \frac { 7 } { t - 1 } , \quad \left( 5 , \frac { 7 } { 4 } \right)$

Accepted Answer

A)  $f ^ {t } ( 5 ) = \frac { 7 } { 16 }$ 
B)  $f ^ { t } ( 5 ) = - \frac { 7 } { 16 }$ 
C)  $f ^ { t } ( 5 ) = - \frac { 7 } { 64 }$ 
D)  $f ^ { t } ( 5 ) = - \frac { 7 } { 4 }$ 
E)  $f ^ { t } ( 5 ) = \frac { 7 } { 4 }$ 
A)  $f ^ {t } ( 5 ) = \frac { 7 } { 16 }$ 
B)  $f ^ { t } ( 5 ) = - \frac { 7 } { 16 }$ 
C)  $f ^ { t } ( 5 ) = - \frac { 7 } { 64 }$ 
D)  $f ^ { t } ( 5 ) = - \frac { 7 } { 4 }$ 
E)  $f ^ { t } ( 5 ) = \frac { 7 } { 4 }$

Question 2

Find an equation of the tangent line to the graph of the function given below at the given point. $7 x ^ { 2 } - 2 x y + 4 y ^ { 2 } - 36 = 0 , \quad ( 2 , - 1 )$ (The coefficients below are given to two decimal places.)

Accepted Answer

A)  $y = 7.50 x + 16.00$ 
B)  $y = 2.50 x + 16.00$ 
C)  $y = 2.50 x - 6.00$ 
D)  $y = - 2.50 x - 6.00$ 
E)  $y = 7.50 x - 16.00$ 
A)  $y = 7.50 x + 16.00$ 
B)  $y = 2.50 x + 16.00$ 
C)  $y = 2.50 x - 6.00$ 
D)  $y = - 2.50 x - 6.00$ 
E)  $y = 7.50 x - 16.00$

Question 3

$\text { Find } \frac { d y } { d x } \text { if } y = e ^ { 4 x ^ { 3 } } \text {. }$

Accepted Answer

A)  $\frac { d y } { d x } = 3 e ^ { 4 x }$ 
B)  $\frac { d y } { d x } = e ^ { 4 x ^ { 3 } }$ 
C)  $\frac { d y } { d x } = 12 x ^ { 2 } \ln \left( 4 ^ { 3 } \right)$ 
D)  $\frac { d y } { d x } = 4 x ^ { 3 } e ^ { 4 x ^ { 3 } - 1 }$ 
E)  $\frac { d y } { d x } = 12 x ^ { 2 } e ^ { 4 x ^ { 3 } }$ 
A)  $\frac { d y } { d x } = 3 e ^ { 4 x }$ 
B)  $\frac { d y } { d x } = e ^ { 4 x ^ { 3 } }$ 
C)  $\frac { d y } { d x } = 12 x ^ { 2 } \ln \left( 4 ^ { 3 } \right)$ 
D)  $\frac { d y } { d x } = 4 x ^ { 3 } e ^ { 4 x ^ { 3 } - 1 }$ 
E)  $\frac { d y } { d x } = 12 x ^ { 2 } e ^ { 4 x ^ { 3 } }$

Question 4

$\frac { d y } { d x } \text { at the point } ( 3,1 ) \text { for the equation } x = y ^ { 3 } - 6 y ^ { 2 } + 8$

Accepted Answer

A)  $- \frac { 1 } { 11 }$ 
B)  $- \frac { 1 } { 7 }$ 
C)  $- \frac { 1 } { 9 }$ 
D)  $\frac { 1 } { 11 }$ 
E)  $\frac { 1 } { 9 }$ 
A)  $- \frac { 1 } { 11 }$ 
B)  $- \frac { 1 } { 7 }$ 
C)  $- \frac { 1 } { 9 }$ 
D)  $\frac { 1 } { 11 }$ 
E)  $\frac { 1 } { 9 }$

Question 5

$\text { Find the derivative of the function } h ( s ) = 7 + \frac { 6 } { 7 } s \text { by the limit process. }$

Accepted Answer

A)  $h ^ { t } ( s ) = 7$ 
B)  $h ^ { t } ( s ) = 7 s + \frac { 6 } { 7 } s ^ { 2 }$ 
C)  $h ^ { t } ( s ) = \frac { 6 } { 7 }$ 
D)  $h ^ { t } ( s ) = \frac { 55 } { 7 }$ 
E)  $h ^ {t } ( s ) = 7 s + \frac { 6 } { 7 }$ 
A)  $h ^ { t } ( s ) = 7$ 
B)  $h ^ { t } ( s ) = 7 s + \frac { 6 } { 7 } s ^ { 2 }$ 
C)  $h ^ { t } ( s ) = \frac { 6 } { 7 }$ 
D)  $h ^ { t } ( s ) = \frac { 55 } { 7 }$ 
E)  $h ^ {t } ( s ) = 7 s + \frac { 6 } { 7 }$

Question 6

Determine all values of , (if any), at which the graph of the function has a horizontal tangent.
$y ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 8$

Accepted Answer

A)  $x = 0$ 
B)  $x = - 8$ 
C)  $x = 0$ and $x = - 8$ 
D)  $x = 0$ and $x = 8$ 
E)  The graph has no horizontal tangents. 
A)  $x = 0$ 
B)  $x = - 8$ 
C)  $x = 0$ and $x = - 8$ 
D)  $x = 0$ and $x = 8$ 
E)  The graph has no horizontal tangents.

Question 7

$\text { Find an equation of the tangent line to the graph of the function } f ( x ) = x ^ { 2 } + 5 x + 2 \text { at }$ the point $( - 5,2 )$

Accepted Answer

A)  $y = - 23$ 
B)  $y = - 5 x - 23$ 
C)  $y = 15 x$ 
D)  $y = 5 x$ 
E)  $y = - 15 x - 73$ 
A)  $y = - 23$ 
B)  $y = - 5 x - 23$ 
C)  $y = 15 x$ 
D)  $y = 5 x$ 
E)  $y = - 15 x - 73$

Question 8

A spherical balloon is inflated with gas at the rate of cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is centimeters?

Accepted Answer

A)  $\frac { d r } { d t } = \frac { 3 } { 98 \pi } \mathrm { cm } / \mathrm { min }$ 
B)  $\frac { d r } { d t } = \frac { 1 } { 98 \pi } \mathrm { cm } / \mathrm { min }$ 
C)  $\frac { d r } { d t } = \frac { 3 } { 196 \pi } \mathrm { cm } / \mathrm { min }$ 
D)  $\frac { d r } { d t } = 98 \pi \mathrm { cm } / \mathrm { min }$ 
E)  $\frac { d r } { d t } = 196 \pi \mathrm { cm } / \mathrm { min }$ 
A)  $\frac { d r } { d t } = \frac { 3 } { 98 \pi } \mathrm { cm } / \mathrm { min }$ 
B)  $\frac { d r } { d t } = \frac { 1 } { 98 \pi } \mathrm { cm } / \mathrm { min }$ 
C)  $\frac { d r } { d t } = \frac { 3 } { 196 \pi } \mathrm { cm } / \mathrm { min }$ 
D)  $\frac { d r } { d t } = 98 \pi \mathrm { cm } / \mathrm { min }$ 
E)  $\frac { d r } { d t } = 196 \pi \mathrm { cm } / \mathrm { min }$

Question 9

Find the derivative of the following function using the limiting process. $$f ( x ) = 3 x ^ { 3 } - 9 x ^ { 2 } - 8$$

Accepted Answer

A)  $f ^ { t } ( x ) = 9 x ^ { 2 } + 18 x$ 
B)  $f ^ { t } ( x ) = 6 x ^ { 2 } - 18 x$ 
C)  $f ^ { t } ( x ) = 9 x ^ { 2 } - 18 x - 8$ 
D)  $f ^ {t } ( x ) = 6 x ^ { 2 } + 18 x$ 
E)  $f ^ { t } ( x ) = 9 x ^ { 2 } - 18 x$ 
A)  $f ^ { t } ( x ) = 9 x ^ { 2 } + 18 x$ 
B)  $f ^ { t } ( x ) = 6 x ^ { 2 } - 18 x$ 
C)  $f ^ { t } ( x ) = 9 x ^ { 2 } - 18 x - 8$ 
D)  $f ^ {t } ( x ) = 6 x ^ { 2 } + 18 x$ 
E)  $f ^ { t } ( x ) = 9 x ^ { 2 } - 18 x$

Question 10

Find the derivative of the following function using the limiting process. $$f ( x ) = \frac { 2 } { x - 3 }$$

Accepted Answer

A)  $f ^ { t } ( x ) = \frac { 2 } { ( x + 3 ) ^ { 2 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 2 x } { ( x - 3 ) ^ { 2 } }$ 
C)  $f ^ { t} ( x ) = - \frac { 2 } { ( x - 3 ) ^ { 2 } }$ 
D)  $f ^ { t } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }$ 
E)  $f ^ { t } ( x ) = - \frac { 2 } { ( x + 3 ) ^ { 2 } }$ 
A)  $f ^ { t } ( x ) = \frac { 2 } { ( x + 3 ) ^ { 2 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 2 x } { ( x - 3 ) ^ { 2 } }$ 
C)  $f ^ { t} ( x ) = - \frac { 2 } { ( x - 3 ) ^ { 2 } }$ 
D)  $f ^ { t } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }$ 
E)  $f ^ { t } ( x ) = - \frac { 2 } { ( x + 3 ) ^ { 2 } }$

Question 11

$\text { Find the derivative of the function } f ( x ) = \frac { e ^ { x } } { 1 + x ^ { 2 } } \text {. Simplify your answer. }$

Accepted Answer

A)  $\frac { - e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
B) $\frac { e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) }$ 
C) $\frac { e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
D) $\frac { e ^ { x } \left( x ^ { 2 } + 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
E)  $\frac { e ^ { x } } { 2 x }$ 
A)  $\frac { - e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
B) $\frac { e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) }$ 
C) $\frac { e ^ { x } \left( x ^ { 2 } - 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
D) $\frac { e ^ { x } \left( x ^ { 2 } + 2 x + 1 \right) } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }$ 
E)  $\frac { e ^ { x } } { 2 x }$

Evaluate the derivative of the function at the given point. $f ( t ) = \frac { 7 } { t - 1 } , \quad \left( 5 , \frac { 7 } { 4 } \right)$

Find an equation of the tangent line to the graph of the function given below at the given point. $7 x ^ { 2 } - 2 x y + 4 y ^ { 2 } - 36 = 0 , \quad ( 2 , - 1 )$ (The coefficients below are given to two decimal places.)

$\text { Find } \frac { d y } { d x } \text { if } y = e ^ { 4 x ^ { 3 } } \text {. }$

$\frac { d y } { d x } \text { at the point } ( 3,1 ) \text { for the equation } x = y ^ { 3 } - 6 y ^ { 2 } + 8$

$\text { Find the derivative of the function } h ( s ) = 7 + \frac { 6 } { 7 } s \text { by the limit process. }$

Determine all values of , (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 8$

$\text { Find an equation of the tangent line to the graph of the function } f ( x ) = x ^ { 2 } + 5 x + 2 \text { at }$ the point $( - 5,2 )$

A spherical balloon is inflated with gas at the rate of cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is centimeters?

Find the derivative of the following function using the limiting process. $f ( x ) = 3 x ^ { 3 } - 9 x ^ { 2 } - 8$

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 2 } { x - 3 }$

$\text { Find the derivative of the function } f ( x ) = \frac { e ^ { x } } { 1 + x ^ { 2 } } \text {. Simplify your answer. }$

Graphs and Models

A Preview of Calculus

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 3: The Derivative and the Tangent Line Problem

Evaluate the derivative of the function at the given point. f(t)=7t−1,(5,74)f ( t ) = \frac { 7 } { t - 1 } , \quad \left( 5 , \frac { 7 } { 4 } \right)f(t)=t−17​,(5,47​)

Find an equation of the tangent line to the graph of the function given below at the given point. 7x2−2xy+4y2−36=0,(2,−1)7 x ^ { 2 } - 2 x y + 4 y ^ { 2 } - 36 = 0 , \quad ( 2 , - 1 )7x2−2xy+4y2−36=0,(2,−1) (The coefficients below are given to two decimal places.)

Find dydx if y=e4x3. \text { Find } \frac { d y } { d x } \text { if } y = e ^ { 4 x ^ { 3 } } \text {. } Find dxdy​ if y=e4x3.

dydx at the point (3,1) for the equation x=y3−6y2+8\frac { d y } { d x } \text { at the point } ( 3,1 ) \text { for the equation } x = y ^ { 3 } - 6 y ^ { 2 } + 8dxdy​ at the point (3,1) for the equation x=y3−6y2+8

Find the derivative of the function h(s)=7+67s by the limit process. \text { Find the derivative of the function } h ( s ) = 7 + \frac { 6 } { 7 } s \text { by the limit process. } Find the derivative of the function h(s)=7+76​s by the limit process.

Determine all values of , (if any), at which the graph of the function has a horizontal tangent. y(x)=x3+12x2+8y ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 8y(x)=x3+12x2+8

A spherical balloon is inflated with gas at the rate of cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is centimeters?

Find the derivative of the following function using the limiting process. f(x)=3x3−9x2−8f ( x ) = 3 x ^ { 3 } - 9 x ^ { 2 } - 8f(x)=3x3−9x2−8

Find the derivative of the following function using the limiting process. f(x)=2x−3f ( x ) = \frac { 2 } { x - 3 }f(x)=x−32​

Find the derivative of the function f(x)=ex1+x2. Simplify your answer. \text { Find the derivative of the function } f ( x ) = \frac { e ^ { x } } { 1 + x ^ { 2 } } \text {. Simplify your answer. } Find the derivative of the function f(x)=1+x2ex​. Simplify your answer.

Graphs and Models

A Preview of Calculus

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Evaluate the derivative of the function at the given point. $f ( t ) = \frac { 7 } { t - 1 } , \quad \left( 5 , \frac { 7 } { 4 } \right)$

Find an equation of the tangent line to the graph of the function given below at the given point. $7 x ^ { 2 } - 2 x y + 4 y ^ { 2 } - 36 = 0 , \quad ( 2 , - 1 )$ (The coefficients below are given to two decimal places.)

$\text { Find } \frac { d y } { d x } \text { if } y = e ^ { 4 x ^ { 3 } } \text {. }$

$\frac { d y } { d x } \text { at the point } ( 3,1 ) \text { for the equation } x = y ^ { 3 } - 6 y ^ { 2 } + 8$

$\text { Find the derivative of the function } h ( s ) = 7 + \frac { 6 } { 7 } s \text { by the limit process. }$

Determine all values of , (if any), at which the graph of the function has a horizontal tangent. $y ( x ) = x ^ { 3 } + 12 x ^ { 2 } + 8$

Find the derivative of the following function using the limiting process. $f ( x ) = 3 x ^ { 3 } - 9 x ^ { 2 } - 8$

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 2 } { x - 3 }$

$\text { Find the derivative of the function } f ( x ) = \frac { e ^ { x } } { 1 + x ^ { 2 } } \text {. Simplify your answer. }$