Question 1

If y = ( $\sqrt [ 3 ] { x }$ + 1 $5 ^ { 5 }$ , find $\frac { d y } { d x }$ .

Accepted Answer

A)  $\frac { 1 } { 3 }$ ( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $x ^ { - 2 / 3 }$ 
B)  $\frac { 5 } { 3 }$ ( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $x ^ { - 2 / 3 }$ 
C)  5( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $\sqrt [ 3 ] { x }$ 
D)  5( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ 
E)  none of these 
A)  $\frac { 1 } { 3 }$ ( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $x ^ { - 2 / 3 }$ 
B)  $\frac { 5 } { 3 }$ ( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $x ^ { - 2 / 3 }$ 
C)  5( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ $\sqrt [ 3 ] { x }$ 
D)  5( $\sqrt [ 3 ] { x }$ + 1 $) ^ { 4 }$ 
E)  none of these B

Question 2

Compute $\frac { d y } { d x }$ using the chain rule where $y = u ^ { 2 }$and u = 3x + 4.
Enter your answer as just a polynomial in x in standard form (no label).

Accepted Answer

18x + 24

Question 3

The cost of manufacturing x units is C dollars, where C = 4x + 6 $\sqrt { x }$ + 5. Weekly production at t weeks from the present is estimated to be x = 2800 + 100t units when t = 8. Find the time rate of change of cost, $\frac { \mathrm { dC } } { \mathrm { dt } }$ . Enter just an integer.

Accepted Answer

405

Question 4

If y = u + 1 - $81 / 3$ and u = $\frac { t } { 6 }$ + 1, find $\frac { d y } { d t }$ .

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  - $\frac { 1 } { 6 t ^ { 2 } }$ ∙ $8 - 2 / 3$ 
C)  - $\frac { t ^ { 2 } } { 6 }$ 
D)  -2 ∙ $8 - 2 / 3$ 
A)  $\frac { 1 } { 6 }$ 
B)  - $\frac { 1 } { 6 t ^ { 2 } }$ ∙ $8 - 2 / 3$ 
C)  - $\frac { t ^ { 2 } } { 6 }$ 
D)  -2 ∙ $8 - 2 / 3$

Question 5

Find the equation of the tangent line to the graph of the function $f ( x ) = \frac { x - 3 } { 3 x - 5 }$ at $x = 1.$

Accepted Answer

A)  y = x 
B)  y = $\frac { 3 } { 2 }$ x - $\frac { 1 } { 2 }$ 
C)  y = -3x + 4 
D)  y = 3x - 2 
A)  y = x 
B)  y = $\frac { 3 } { 2 }$ x - $\frac { 1 } { 2 }$ 
C)  y = -3x + 4 
D)  y = 3x - 2

Question 6

Find the slope of the tangent line to f(x) = $\frac { x ^ { 2 } - 1 } { x + 2 }$ at (-1, f(-1)).
Enter just an integer.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 7

Mr. Smith is 6 ft tall and walks at a constant rate of 2 ft/sec toward a street light that is 10 ft above the ground. At what rate is the length of his shadow changing when he is 6 ft from the base of the pole that supports the light? Enter just an integer (no units).

Accepted Answer

The answer of Mr. Smith is 6 ft tall and...

Question 8

The radius, r, of a sphere is increasing. For what value of r is $\frac { \mathrm { dV } } { \mathrm { dt } }$ equal to 64π times the rate of increase of r. Enter just an integer.

Accepted Answer

The answer of The radius, r, of a sphere is...

Question 9

Find the slope of the tangent line to f(x) = $\frac { x } { x ^ { 2 } + 1 }$ at (2, f(2)).
Enter just a reduced fraction.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 10

Find the slope of the tangent line to f(x) = 4( $x ^ { 3 }$ + 1)(2 $x ^ { 2 }$ + 2x + 1 $) ^ { 4 }$ at (-1, f(-1)).
Enter just an integer.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 11

Suppose that x and y are related by the equation $\frac { x ^ { 2 } } { 4 }$ + $\frac { y ^ { 3 } } { 2 }$ = 4. Use implicit differentiation to determine $\frac { d y } { d x }$ .

Accepted Answer

A)  $\frac { d y } { d x }$ = - $\frac { 2 x } { y ^ { 3 } }$ 
B)  $\frac { d y } { d x }$ = $\frac { 2 y ^ { 2 } } { x }$ 
C)  $\frac { d y } { d x }$ = $\frac { 1 } { 3 y ^ { 2 } }$ , y ≠ 0 
D)  $\frac { d y } { d x }$ = - $\frac { x } { 3 y ^ { 2 } }$ , y ≠ 0 
A)  $\frac { d y } { d x }$ = - $\frac { 2 x } { y ^ { 3 } }$ 
B)  $\frac { d y } { d x }$ = $\frac { 2 y ^ { 2 } } { x }$ 
C)  $\frac { d y } { d x }$ = $\frac { 1 } { 3 y ^ { 2 } }$ , y ≠ 0 
D)  $\frac { d y } { d x }$ = - $\frac { x } { 3 y ^ { 2 } }$ , y ≠ 0

Question 12

Differentiate
-g(x) = ( $x ^ { 3 }$ + 1)(3 $x ^ { 2 }$ - 1)

Accepted Answer

A)  -3 $x ^ { 4 }$ + 3 $x ^ { 2 }$ + 6x 
B)  15 $x ^ { 4 }$ - 3 $x ^ { 2 }$ + 6x 
C)  -3 $x ^ { 4 }$ - 3 $x ^ { 2 }$ + 6x 
D)  3 $x ^ { 4 }$ - 3 $x ^ { 2 }$ - 6x 
A)  -3 $x ^ { 4 }$ + 3 $x ^ { 2 }$ + 6x 
B)  15 $x ^ { 4 }$ - 3 $x ^ { 2 }$ + 6x 
C)  -3 $x ^ { 4 }$ - 3 $x ^ { 2 }$ + 6x 
D)  3 $x ^ { 4 }$ - 3 $x ^ { 2 }$ - 6x

Question 13

One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x), in degrees Celsius, is given approximately by: T(x) = $\frac { 5 x ^ { 2 } } { 9 }$ $\left. 1 - \frac { x } { 9 } \right)$ - $\frac { 160 } { 9 }$ , 0 ≤ x ≤ 6. Find the sensitivity, T'(x), of the body to a dosage of three milligrams.

Accepted Answer

A)  $\frac { 5 } { 3 }$ degrees per milligram 
B)  - $\frac { 10 } { 9 }$ degrees per milligram 
C)  - $\frac { 5 } { 3 }$ degrees per milligram 
D)  - $\frac { 10 } { 27 }$ degree per milligram 
A)  $\frac { 5 } { 3 }$ degrees per milligram 
B)  - $\frac { 10 } { 9 }$ degrees per milligram 
C)  - $\frac { 5 } { 3 }$ degrees per milligram 
D)  - $\frac { 10 } { 27 }$ degree per milligram

Question 14

Use implicit differentiation to determine $\frac { d y } { d x }$ where xy + 10 = $x ^ { 2 }$ . Is $\frac { d y } { d x } = \frac { 2 x - y } { x }$ correct?

Accepted Answer

A) True 
 B)False

Question 15

Find the slope of the tangent line to the graph of y = $\frac { x } { \sqrt { \left( 8 - x ^ { 2 } \right) } }$ at the point (2, 1).
Enter just an integer.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 16

Use implicit differentiation to determine $\frac{d y}{d x}$
 where $x ^ { 2 }$ + $y ^ { 2 }$ = 4. Is $\frac { d y } { d x } = \frac { x } { y }$ correct?

Accepted Answer

A) True 
 B)False

Question 17

Let f(x) = $x ^ { 3 }$ . Using the chain rule, find an expression for the derivative of [f(g(x))].

Accepted Answer

A)  3 $x ^ { 2 }$ g'(x $) ^ { 3 }$ 
B)  3[g(x) $] ^ { 2 }$ 
C)  g'(x $) ^ { 3 }$ 
D)  3[g(x) $] ^ { 2 }$ g'(x) 
E)  none of these 
A)  3 $x ^ { 2 }$ g'(x $) ^ { 3 }$ 
B)  3[g(x) $] ^ { 2 }$ 
C)  g'(x $) ^ { 3 }$ 
D)  3[g(x) $] ^ { 2 }$ g'(x) 
E)  none of these

Question 18

Determine the rate of change of $\sqrt { 2 x + 4 }$ with respect to x at x = 1.
Enter just a reduced quotient of form $\frac { a } { b }$ .

Accepted Answer

The answer of Determine the rate of change of \(\sqrt...

Question 19

Differentiate: f(x) = (( $x ^ { 5 }$ + 2 $) ^ { 3 }$ + 1 $) ^ { 4 }$ at x = -1.
Enter just an integer.

Accepted Answer

The answer of Differentiate: f(x) = (( \(x ^ {...

Question 20

Differentiate
-f(x) = $\frac { 2 x - 7 } { 3 x - 2 }$

Accepted Answer

A)  - $\frac { 17 } { ( 2 x - 7 ) ^ { 2 } }$ 
B)  $\frac { 17 } { ( 2 x - 7 ) ^ { 2 } }$ 
C)  - $\frac { 17 } { ( 3 x - 2 ) ^ { 2 } }$ 
D)  $\frac { 17 } { ( 3 x - 2 ) ^ { 2 } }$ 
A)  - $\frac { 17 } { ( 2 x - 7 ) ^ { 2 } }$ 
B)  $\frac { 17 } { ( 2 x - 7 ) ^ { 2 } }$ 
C)  - $\frac { 17 } { ( 3 x - 2 ) ^ { 2 } }$ 
D)  $\frac { 17 } { ( 3 x - 2 ) ^ { 2 } }$

If y = ( $\sqrt [ 3 ] { x }$ + 1 $5 ^ { 5 }$ , find $\frac { d y } { d x }$ .

Compute $\frac { d y } { d x }$ using the chain rule where $y = u ^ { 2 }$ and u = 3x + 4. Enter your answer as just a polynomial in x in standard form (no label).

The cost of manufacturing x units is C dollars, where C = 4x + 6 $\sqrt { x }$ + 5. Weekly production at t weeks from the present is estimated to be x = 2800 + 100t units when t = 8. Find the time rate of change of cost, $\frac { \mathrm { dC } } { \mathrm { dt } }$ . Enter just an integer.

If y = u + 1 - $81 / 3$ and u = $\frac { t } { 6 }$ + 1, find $\frac { d y } { d t }$ .

Find the equation of the tangent line to the graph of the function $f ( x ) = \frac { x - 3 } { 3 x - 5 }$ at $x = 1.$

Find the slope of the tangent line to f(x) = $\frac { x ^ { 2 } - 1 } { x + 2 }$ at (-1, f(-1)). Enter just an integer.

Mr. Smith is 6 ft tall and walks at a constant rate of 2 ft/sec toward a street light that is 10 ft above the ground. At what rate is the length of his shadow changing when he is 6 ft from the base of the pole that supports the light? Enter just an integer (no units).

The radius, r, of a sphere is increasing. For what value of r is $\frac { \mathrm { dV } } { \mathrm { dt } }$ equal to 64π times the rate of increase of r. Enter just an integer.

Find the slope of the tangent line to f(x) = $\frac { x } { x ^ { 2 } + 1 }$ at (2, f(2)). Enter just a reduced fraction.

Find the slope of the tangent line to f(x) = 4( $x ^ { 3 }$ + 1)(2 $x ^ { 2 }$ + 2x + 1 $) ^ { 4 }$ at (-1, f(-1)). Enter just an integer.

Suppose that x and y are related by the equation $\frac { x ^ { 2 } } { 4 }$ + $\frac { y ^ { 3 } } { 2 }$ = 4. Use implicit differentiation to determine $\frac { d y } { d x }$ .

Differentiate -g(x) = ( $x ^ { 3 }$ + 1)(3 $x ^ { 2 }$ - 1)

Use implicit differentiation to determine $\frac { d y } { d x }$ where xy + 10 = $x ^ { 2 }$ . Is $\frac { d y } { d x } = \frac { 2 x - y } { x }$ correct?

Find the slope of the tangent line to the graph of y = $\frac { x } { \sqrt { \left( 8 - x ^ { 2 } \right) } }$ at the point (2, 1). Enter just an integer.

Use implicit differentiation to determine $\frac{d y}{d x}$ where $x ^ { 2 }$ + $y ^ { 2 }$ = 4. Is $\frac { d y } { d x } = \frac { x } { y }$ correct?

Let f(x) = $x ^ { 3 }$ . Using the chain rule, find an expression for the derivative of [f(g(x))].

Determine the rate of change of $\sqrt { 2 x + 4 }$ with respect to x at x = 1. Enter just a reduced quotient of form $\frac { a } { b }$ .

Differentiate: f(x) = (( $x ^ { 5 }$ + 2 $) ^ { 3 }$ + 1 $) ^ { 4 }$ at x = -1. Enter just an integer.

Differentiate -f(x) = $\frac { 2 x - 7 } { 3 x - 2 }$

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 3: Techniques of Differentiation

If y = ( x3\sqrt [ 3 ] { x }3x​ + 1 555 ^ { 5 }55 , find dydx\frac { d y } { d x }dxdy​ .

Compute dydx\frac { d y } { d x }dxdy​ using the chain rule where y=u2y = u ^ { 2 }y=u2 and u = 3x + 4. Enter your answer as just a polynomial in x in standard form (no label).

If y = u + 1 - 81/381 / 381/3 and u = t6\frac { t } { 6 }6t​ + 1, find dydt\frac { d y } { d t }dtdy​ .

Find the equation of the tangent line to the graph of the function f(x)=x−33x−5f ( x ) = \frac { x - 3 } { 3 x - 5 }f(x)=3x−5x−3​ at x=1.x = 1.x=1.

Find the slope of the tangent line to f(x) = x2−1x+2\frac { x ^ { 2 } - 1 } { x + 2 }x+2x2−1​ at (-1, f(-1)). Enter just an integer.

Mr. Smith is 6 ft tall and walks at a constant rate of 2 ft/sec toward a street light that is 10 ft above the ground. At what rate is the length of his shadow changing when he is 6 ft from the base of the pole that supports the light? Enter just an integer (no units).

The radius, r, of a sphere is increasing. For what value of r is dVdt\frac { \mathrm { dV } } { \mathrm { dt } }dtdV​ equal to 64π times the rate of increase of r. Enter just an integer.

Find the slope of the tangent line to f(x) = xx2+1\frac { x } { x ^ { 2 } + 1 }x2+1x​ at (2, f(2)). Enter just a reduced fraction.

Find the slope of the tangent line to f(x) = 4( x3x ^ { 3 }x3 + 1)(2 x2x ^ { 2 }x2 + 2x + 1 )4) ^ { 4 })4 at (-1, f(-1)). Enter just an integer.

Suppose that x and y are related by the equation x24\frac { x ^ { 2 } } { 4 }4x2​ + y32\frac { y ^ { 3 } } { 2 }2y3​ = 4. Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ .

Differentiate -g(x) = ( x3x ^ { 3 }x3 + 1)(3 x2x ^ { 2 }x2 - 1)

Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ where xy + 10 = x2x ^ { 2 }x2 . Is dydx=2x−yx\frac { d y } { d x } = \frac { 2 x - y } { x }dxdy​=x2x−y​ correct?

Find the slope of the tangent line to the graph of y = x(8−x2)\frac { x } { \sqrt { \left( 8 - x ^ { 2 } \right) } }(8−x2)​x​ at the point (2, 1). Enter just an integer.

Use implicit differentiation to determine dydx\frac{d y}{d x}dxdy​ where x2x ^ { 2 }x2 + y2y ^ { 2 }y2 = 4. Is dydx=xy\frac { d y } { d x } = \frac { x } { y }dxdy​=yx​ correct?

Let f(x) = x3x ^ { 3 }x3 . Using the chain rule, find an expression for the derivative of [f(g(x))].

Determine the rate of change of 2x+4\sqrt { 2 x + 4 }2x+4​ with respect to x at x = 1. Enter just a reduced quotient of form ab\frac { a } { b }ba​ .

Differentiate: f(x) = (( x5x ^ { 5 }x5 + 2 )3) ^ { 3 })3 + 1 )4) ^ { 4 })4 at x = -1. Enter just an integer.

Differentiate -f(x) = 2x−73x−2\frac { 2 x - 7 } { 3 x - 2 }3x−22x−7​

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

If y = ( $\sqrt [ 3 ] { x }$ + 1 $5 ^ { 5 }$ , find $\frac { d y } { d x }$ .

Compute $\frac { d y } { d x }$ using the chain rule where $y = u ^ { 2 }$ and u = 3x + 4. Enter your answer as just a polynomial in x in standard form (no label).

If y = u + 1 - $81 / 3$ and u = $\frac { t } { 6 }$ + 1, find $\frac { d y } { d t }$ .

Find the equation of the tangent line to the graph of the function $f ( x ) = \frac { x - 3 } { 3 x - 5 }$ at $x = 1.$

Find the slope of the tangent line to f(x) = $\frac { x ^ { 2 } - 1 } { x + 2 }$ at (-1, f(-1)). Enter just an integer.

The radius, r, of a sphere is increasing. For what value of r is $\frac { \mathrm { dV } } { \mathrm { dt } }$ equal to 64π times the rate of increase of r. Enter just an integer.

Find the slope of the tangent line to f(x) = $\frac { x } { x ^ { 2 } + 1 }$ at (2, f(2)). Enter just a reduced fraction.

Find the slope of the tangent line to f(x) = 4( $x ^ { 3 }$ + 1)(2 $x ^ { 2 }$ + 2x + 1 $) ^ { 4 }$ at (-1, f(-1)). Enter just an integer.

Suppose that x and y are related by the equation $\frac { x ^ { 2 } } { 4 }$ + $\frac { y ^ { 3 } } { 2 }$ = 4. Use implicit differentiation to determine $\frac { d y } { d x }$ .

Differentiate -g(x) = ( $x ^ { 3 }$ + 1)(3 $x ^ { 2 }$ - 1)

Use implicit differentiation to determine $\frac { d y } { d x }$ where xy + 10 = $x ^ { 2 }$ . Is $\frac { d y } { d x } = \frac { 2 x - y } { x }$ correct?

Find the slope of the tangent line to the graph of y = $\frac { x } { \sqrt { \left( 8 - x ^ { 2 } \right) } }$ at the point (2, 1). Enter just an integer.

Use implicit differentiation to determine $\frac{d y}{d x}$ where $x ^ { 2 }$ + $y ^ { 2 }$ = 4. Is $\frac { d y } { d x } = \frac { x } { y }$ correct?

Let f(x) = $x ^ { 3 }$ . Using the chain rule, find an expression for the derivative of [f(g(x))].

Determine the rate of change of $\sqrt { 2 x + 4 }$ with respect to x at x = 1. Enter just a reduced quotient of form $\frac { a } { b }$ .

Differentiate: f(x) = (( $x ^ { 5 }$ + 2 $) ^ { 3 }$ + 1 $) ^ { 4 }$ at x = -1. Enter just an integer.

Differentiate -f(x) = $\frac { 2 x - 7 } { 3 x - 2 }$