Question 1

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$ .

Accepted Answer

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

Question 2

Differentiate: $f ( x ) = \sqrt { x } + \frac { 1 } { \sqrt { x } }$

Accepted Answer

A)  0 
B)  x 
C)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 1 } { 2 \sqrt { x ^ { 3 } } }$ 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { 2 \sqrt { x ^ { 3 } } }$ 
A)  0 
B)  x 
C)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 1 } { 2 \sqrt { x ^ { 3 } } }$ 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { 2 \sqrt { x ^ { 3 } } }$

Question 3

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

Accepted Answer

A)  $80 x ^ { 3 }$ 
B)  $- \frac { 80 } { 11 } x ^ { 3 }$ 
C)  $60 x ^ { 2 } + 11$ 
D)  $60 x ^ { 2 } - 100$ 
A)  $80 x ^ { 3 }$ 
B)  $- \frac { 80 } { 11 } x ^ { 3 }$ 
C)  $60 x ^ { 2 } + 11$ 
D)  $60 x ^ { 2 } - 100$

Question 4

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at the point $\left( 6 , \frac { 1 } { 6 } \right)$ .

Accepted Answer

A)  $y = - \frac { 1 } { 36 } x + \frac { 1 } { 3 }$ 
B)  $y = - \frac { 1 } { 6 } x + \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 36 } x$ 
D)  $y = \frac { 1 } { 6 } x + \frac { 1 } { 3 }$ 
A)  $y = - \frac { 1 } { 36 } x + \frac { 1 } { 3 }$ 
B)  $y = - \frac { 1 } { 6 } x + \frac { 1 } { 3 }$ 
C)  $y = \frac { 1 } { 36 } x$ 
D)  $y = \frac { 1 } { 6 } x + \frac { 1 } { 3 }$

Question 5

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 2 t - 1$ . What is the velocity of the object as a function of t?

Accepted Answer

A)  $3 t ^ { 2 }$ 
B)  $3 t ^ { 2 } + 2$ 
C)  3 
D)  2 
A)  $3 t ^ { 2 }$ 
B)  $3 t ^ { 2 } + 2$ 
C)  3 
D)  2

Question 6

At a certain factory, the total cost of manufacturing q units during the daily production run is $C ( q ) = 0.3 q ^ { 2 } + 0.8 q + 800$ dollars. It has been determined that approximately $q ( t ) = t ^ { 2 } + 80 t$ units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with respect to time 2 hours after production begins.

Accepted Answer

It is incr

Question 7

If $x ^ { 3 } + y ^ { 3 } = x + y$ , then $\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }$ .

Accepted Answer

A) True 
 B)False

Question 8

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 5 } y ^ { 5 } - 5 x y = 6 x + y - 8$ ; (0, 8)

Accepted Answer

A)  $y = - \frac { 1 } { 46 } x + 8$ 
B)  y = -46x + 8 
C)  y = 46x + 8 
D)  $y = \frac { 1 } { 46 } x + 8$ 
A)  $y = - \frac { 1 } { 46 } x + 8$ 
B)  y = -46x + 8 
C)  y = 46x + 8 
D)  $y = \frac { 1 } { 46 } x + 8$

Question 9

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 3 }$ with respect to t when t = 3?

Accepted Answer

The answer of What is the rate of change of...

Question 10

An environmental study of a certain suburban community suggests that t years from now the average level of carbon monoxide in the air will be $Q ( t ) = 0.07 t ^ { 2 } + 0.2 t + 2.8$ ppm. The rate that the carbon monoxide level will change with respect to time 2 years from now will be 0.048 ppm/yr.

Accepted Answer

A) True 
 B)False

Question 11

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

Accepted Answer

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

Question 12

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 3 t - 7$ . What is the acceleration of the object as a function of time?

Accepted Answer

A)  2 
B)  2t + 3 
C)  2t 
D)  t 
A)  2 
B)  2t + 3 
C)  2t 
D)  t

Question 13

Differentiating $f ( x ) = \frac { 1 } { 5 } x ^ { 6 } - 3 x ^ { 4 } + 2 x - 4$ gives $\frac { 6 } { 5 } x ^ { 5 } - 12 x ^ { 3 } + 2$ .

Accepted Answer

A) True 
 B)False

Question 14

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 3 u - 1 }$ and $u = \frac { 1 } { x + 2 }$ .

Accepted Answer

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

Question 15

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Accepted Answer

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

Question 16

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 7 u - 1 }$ and $u = \frac { 1 } { x + 3 }$ .

Accepted Answer

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

Question 17

If $f ( x ) = \sqrt { 1 - 3 x ^ { 2 } }$ , then $f ^ { \prime \prime } ( x ) = \frac { - 3 } { \left( 1 - 3 x ^ { 2 } \right) ^ { 3 / 2 } }$ .

Accepted Answer

A) True 
 B)False

Question 18

If an object moves in such a way that after t seconds, the distance from its starting point is $D ( t ) = t ^ { 3 } - 15 t ^ { 2 } + 80 t$ meters, find the acceleration after 2 seconds in meters/s2.

Accepted Answer

Question 19

The equation of the line that is tangent to the curve $f ( x ) = \left( 3 x ^ { 5 } - 7 x ^ { 2 } + 5 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -5) is y = 5x - 5.

Accepted Answer

A) True 
 B)False

Question 20

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 50 - \frac { 7 } { 2 t + 1 }$ thousand people. At what rate, in people/year will the population be growing 3 years from now?

Accepted Answer

The answer of It is estimated that t years from...

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$ .

Differentiate: $f ( x ) = \sqrt { x } + \frac { 1 } { \sqrt { x } }$

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at the point $\left( 6 , \frac { 1 } { 6 } \right)$ .

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 2 t - 1$ . What is the velocity of the object as a function of t?

If $x ^ { 3 } + y ^ { 3 } = x + y$ , then $\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }$ .

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 5 } y ^ { 5 } - 5 x y = 6 x + y - 8$ ; (0, 8)

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 3 }$ with respect to t when t = 3?

An environmental study of a certain suburban community suggests that t years from now the average level of carbon monoxide in the air will be $Q ( t ) = 0.07 t ^ { 2 } + 0.2 t + 2.8$ ppm. The rate that the carbon monoxide level will change with respect to time 2 years from now will be 0.048 ppm/yr.

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 3 t - 7$ . What is the acceleration of the object as a function of time?

Differentiating $f ( x ) = \frac { 1 } { 5 } x ^ { 6 } - 3 x ^ { 4 } + 2 x - 4$ gives $\frac { 6 } { 5 } x ^ { 5 } - 12 x ^ { 3 } + 2$ .

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 3 u - 1 }$ and $u = \frac { 1 } { x + 2 }$ .

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 7 u - 1 }$ and $u = \frac { 1 } { x + 3 }$ .

If $f ( x ) = \sqrt { 1 - 3 x ^ { 2 } }$ , then $f ^ { \prime \prime } ( x ) = \frac { - 3 } { \left( 1 - 3 x ^ { 2 } \right) ^ { 3 / 2 } }$ .

If an object moves in such a way that after t seconds, the distance from its starting point is $D ( t ) = t ^ { 3 } - 15 t ^ { 2 } + 80 t$ meters, find the acceleration after 2 seconds in meters/s².

The equation of the line that is tangent to the curve $f ( x ) = \left( 3 x ^ { 5 } - 7 x ^ { 2 } + 5 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -5) is y = 5x - 5.

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 50 - \frac { 7 } { 2 t + 1 }$ thousand people. At what rate, in people/year will the population be growing 3 years from now?

Functions, Graphs, and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 2: Differentiation: Basic Concepts

Find dydx\frac { d y } { d x }dxdy​ if y=u3y = \sqrt [ 3 ] { u }y=3u​ and u=x4−3x3−7u = x ^ { 4 } - 3 x ^ { 3 } - 7u=x4−3x3−7 .

Differentiate: f(x)=x+1xf ( x ) = \sqrt { x } + \frac { 1 } { \sqrt { x } }f(x)=x​+x​1​

Use implicit differentiation to find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ for 4x5+11y=1004 x ^ { 5 } + 11 y = 1004x5+11y=100 .

Find the equation of the tangent line to the graph of f(x)=1xf ( x ) = \frac { 1 } { x }f(x)=x1​ at the point (6,16)\left( 6 , \frac { 1 } { 6 } \right)(6,61​) .

The displacement function of a moving object is described by s(t)=t3+2t−1s ( t ) = t ^ { 3 } + 2 t - 1s(t)=t3+2t−1 . What is the velocity of the object as a function of t?

If x3+y3=x+yx ^ { 3 } + y ^ { 3 } = x + yx3+y3=x+y , then dydx=3x2−13y2−1\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }dxdy​=3y2−13x2−1​ .

Find the equation of the tangent line to the given curve at the specified point. x5y5−5xy=6x+y−8x ^ { 5 } y ^ { 5 } - 5 x y = 6 x + y - 8x5y5−5xy=6x+y−8 ; (0, 8)

What is the rate of change of f(t)=4t−5t+3f ( t ) = \frac { 4 t - 5 } { t + 3 }f(t)=t+34t−5​ with respect to t when t = 3?

The equation of the line tangent to the graph of f(x)=3xf ( x ) = 3 \sqrt { x }f(x)=3x​ at x = 1 is

The displacement function of a moving object is described by s(t)=t2+3t−7s ( t ) = t ^ { 2 } + 3 t - 7s(t)=t2+3t−7 . What is the acceleration of the object as a function of time?

Differentiating f(x)=15x6−3x4+2x−4f ( x ) = \frac { 1 } { 5 } x ^ { 6 } - 3 x ^ { 4 } + 2 x - 4f(x)=51​x6−3x4+2x−4 gives 65x5−12x3+2\frac { 6 } { 5 } x ^ { 5 } - 12 x ^ { 3 } + 256​x5−12x3+2 .

Find dydx\frac { d y } { d x }dxdy​ if y=13u−1y = \frac { 1 } { 3 u - 1 }y=3u−11​ and u=1x+2u = \frac { 1 } { x + 2 }u=x+21​ .

Find the equation of the tangent line to the curve f(x)=4x−xf ( x ) = \frac { 4 } { x } - xf(x)=x4​−x at the point where x = 1.

Find dydx\frac { d y } { d x }dxdy​ if y=17u−1y = \frac { 1 } { 7 u - 1 }y=7u−11​ and u=1x+3u = \frac { 1 } { x + 3 }u=x+31​ .

If f(x)=1−3x2f ( x ) = \sqrt { 1 - 3 x ^ { 2 } }f(x)=1−3x2​ , then f′′(x)=−3(1−3x2)3/2f ^ { \prime \prime } ( x ) = \frac { - 3 } { \left( 1 - 3 x ^ { 2 } \right) ^ { 3 / 2 } }f′′(x)=(1−3x2)3/2−3​ .

If an object moves in such a way that after t seconds, the distance from its starting point is D(t)=t3−15t2+80tD ( t ) = t ^ { 3 } - 15 t ^ { 2 } + 80 tD(t)=t3−15t2+80t meters, find the acceleration after 2 seconds in meters/s2.

The equation of the line that is tangent to the curve f(x)=(3x5−7x2+5)(x3+x−1)f ( x ) = \left( 3 x ^ { 5 } - 7 x ^ { 2 } + 5 \right) \left( x ^ { 3 } + x - 1 \right)f(x)=(3x5−7x2+5)(x3+x−1) at the point (0, -5) is y = 5x - 5.

It is estimated that t years from now, the population of a certain suburban community will be p(t)=50−72t+1p ( t ) = 50 - \frac { 7 } { 2 t + 1 }p(t)=50−2t+17​ thousand people. At what rate, in people/year will the population be growing 3 years from now?

Functions, Graphs, and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$ .

Differentiate: $f ( x ) = \sqrt { x } + \frac { 1 } { \sqrt { x } }$

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at the point $\left( 6 , \frac { 1 } { 6 } \right)$ .

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 2 t - 1$ . What is the velocity of the object as a function of t?

If $x ^ { 3 } + y ^ { 3 } = x + y$ , then $\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }$ .

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 5 } y ^ { 5 } - 5 x y = 6 x + y - 8$ ; (0, 8)

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 3 }$ with respect to t when t = 3?

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 3 t - 7$ . What is the acceleration of the object as a function of time?

Differentiating $f ( x ) = \frac { 1 } { 5 } x ^ { 6 } - 3 x ^ { 4 } + 2 x - 4$ gives $\frac { 6 } { 5 } x ^ { 5 } - 12 x ^ { 3 } + 2$ .

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 3 u - 1 }$ and $u = \frac { 1 } { x + 2 }$ .

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Find $\frac { d y } { d x }$ if $y = \frac { 1 } { 7 u - 1 }$ and $u = \frac { 1 } { x + 3 }$ .

If $f ( x ) = \sqrt { 1 - 3 x ^ { 2 } }$ , then $f ^ { \prime \prime } ( x ) = \frac { - 3 } { \left( 1 - 3 x ^ { 2 } \right) ^ { 3 / 2 } }$ .

If an object moves in such a way that after t seconds, the distance from its starting point is $D ( t ) = t ^ { 3 } - 15 t ^ { 2 } + 80 t$ meters, find the acceleration after 2 seconds in meters/s².

The equation of the line that is tangent to the curve $f ( x ) = \left( 3 x ^ { 5 } - 7 x ^ { 2 } + 5 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -5) is y = 5x - 5.

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 50 - \frac { 7 } { 2 t + 1 }$ thousand people. At what rate, in people/year will the population be growing 3 years from now?