Question 1

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (4, 17).

Accepted Answer

A)  y = 8x - 15 
B)  Not defined 
C)  y = 17 
D)  x = 4 
A)  y = 8x - 15 
B)  Not defined 
C)  y = 17 
D)  x = 4

Question 2

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 7 u ^ { 2 } - 6$ and $u = x ^ { 2 } + x - 8$ .

Accepted Answer

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

Question 3

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 8 x$ at x = 7 is

Accepted Answer

A)  y = 22x - 49 
B)  y = 22x - 2,744 
C)  y = 22x - 7 
D)  y = 22x - 343 
A)  y = 22x - 49 
B)  y = 22x - 2,744 
C)  y = 22x - 7 
D)  y = 22x - 343

Question 4

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Accepted Answer

A)  $\frac { 9 } { 4 }$ 
B)  4 
C)  $\frac { 19 } { 16 }$ 
D)  $\frac { 17 } { 4 }$ 
A)  $\frac { 9 } { 4 }$ 
B)  4 
C)  $\frac { 19 } { 16 }$ 
D)  $\frac { 17 } { 4 }$

Question 5

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

Accepted Answer

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

Question 6

If the position of an object moving along a straight line is given by $s ( t ) = t ^ { 3 } - 7 t ^ { 2 } + 8 t$ at time t, find the object's velocity as a function of time.

Accepted Answer

A)  $v ( t ) = 3 t ^ { 2 } - 7 t + 8$ 
B)  $v ( t ) = t ^ { 2 } - 14 t$ 
C)  $v ( t ) = t ^ { 2 } - 7 t + 8$ 
D)  $v ( t ) = 3 t ^ { 2 } - 14 t + 8$ 
A)  $v ( t ) = 3 t ^ { 2 } - 7 t + 8$ 
B)  $v ( t ) = t ^ { 2 } - 14 t$ 
C)  $v ( t ) = t ^ { 2 } - 7 t + 8$ 
D)  $v ( t ) = 3 t ^ { 2 } - 14 t + 8$

Question 7

An appliance store manager estimates that for x television ads run per day, $R ( x ) = - 0.01 x ^ { 3 } + x ^ { 2 } - 3 x + 200$ refrigerators will be sold per month. Find $R ^ { \prime } ( 4 )$ and interpret what it tells us about sales.

Accepted Answer

A)  $R ^ { \prime } ( 4 ) = 4.52$ sales will be increasing at about 5 refrigerators per month per ad when they're running 4 ads. 
B)  $R ^ { \prime } ( 4 ) = 203.36$ the cost of refrigerators will be rising by $203.36 if they're selling 4 per day. 
C)  $R ^ { \prime } ( 4 ) = 4.52 \text {; }$ they'll sell about 5 refrigerators if they run 4 ads per day. 
D)  $R ^ { \prime } ( 4 ) = 203.36 \text {; }$ they'll sell about 203 refrigerators if they run 4 ads per day. 
A)  $R ^ { \prime } ( 4 ) = 4.52$ sales will be increasing at about 5 refrigerators per month per ad when they're running 4 ads. 
B)  $R ^ { \prime } ( 4 ) = 203.36$ the cost of refrigerators will be rising by $203.36 if they're selling 4 per day. 
C)  $R ^ { \prime } ( 4 ) = 4.52 \text {; }$ they'll sell about 5 refrigerators if they run 4 ads per day. 
D)  $R ^ { \prime } ( 4 ) = 203.36 \text {; }$ they'll sell about 203 refrigerators if they run 4 ads per day.

Question 8

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

Accepted Answer

The answer of If the total cost of manufacturing q...

Question 9

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

Accepted Answer

A) True 
 B)False

Question 10

Find $\frac { d y } { d x }$ , where $\frac { 3 } { x } + \frac { 1 } { 2 y } = 5$ .

Accepted Answer

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

Question 11

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 is

Accepted Answer

A)  y = 16x - 7 
B)  y = 16x - 49 
C)  y = 16x - 343 
D)  y = 16x - 686 
A)  y = 16x - 7 
B)  y = 16x - 49 
C)  y = 16x - 343 
D)  y = 16x - 686

Question 12

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 5 t - 4$ . What is the velocity of the object as a function of time?

Accepted Answer

A)  $3 t ^ { 2 } + 5$ 
B)  $3 t ^ { 2 }$ 
C)  3 
D)  2 
A)  $3 t ^ { 2 } + 5$ 
B)  $3 t ^ { 2 }$ 
C)  3 
D)  2

Question 13

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 20 - \frac { 2 } { 4 t + 3 }$ thousand people. At what rate will the population be growing 3 years from now?

Accepted Answer

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

Question 14

If $f ( x ) = \frac { 3 x - 1 } { x + 1 }$ , what is $f ^ { \prime } ( x )$ ?

Accepted Answer

The answer of If \(f ( x ) = \frac...

Question 15

Find an equation for the tangent line to the curve $y = \sqrt { 2 + \frac { 1 } { 5 } x }$ at the point where x = -1.

Accepted Answer

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

Question 16

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

Accepted Answer

A)  7 
B)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
C)  0 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
A)  7 
B)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
C)  0 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$

Question 17

An equation for the tangent line to the curve $f ( x ) = \sqrt { 3 x ^ { 2 } + 5 x }$ at the point where x = 1 is $y = 2 x - 1$ .

Accepted Answer

A) True 
 B)False

Question 18

Find the equation of the tangent line to the curve $f ( x ) = x ^ { 3 } - x ^ { 2 } + 1$ at the point (1, 1).

Accepted Answer

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

Question 19

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) =5x3 - x2 + 5

Accepted Answer

A)  13 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 9 } { 13 }$ 
D)  $\frac { 13 } { 9 }$ 
A)  13 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 9 } { 13 }$ 
D)  $\frac { 13 } { 9 }$

Question 20

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$ .

Accepted Answer

A) True 
 B)False

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (4, 17).

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 7 u ^ { 2 } - 6$ and $u = x ^ { 2 } + x - 8$ .

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 8 x$ at x = 7 is

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

If the position of an object moving along a straight line is given by $s ( t ) = t ^ { 3 } - 7 t ^ { 2 } + 8 t$ at time t, find the object's velocity as a function of time.

An appliance store manager estimates that for x television ads run per day, $R ( x ) = - 0.01 x ^ { 3 } + x ^ { 2 } - 3 x + 200$ refrigerators will be sold per month. Find $R ^ { \prime } ( 4 )$ and interpret what it tells us about sales.

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

Find $\frac { d y } { d x }$ , where $\frac { 3 } { x } + \frac { 1 } { 2 y } = 5$ .

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 is

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 5 t - 4$ . What is the velocity of the object as a function of time?

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 20 - \frac { 2 } { 4 t + 3 }$ thousand people. At what rate will the population be growing 3 years from now?

If $f ( x ) = \frac { 3 x - 1 } { x + 1 }$ , what is $f ^ { \prime } ( x )$ ?

Find an equation for the tangent line to the curve $y = \sqrt { 2 + \frac { 1 } { 5 } x }$ at the point where x = -1.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

An equation for the tangent line to the curve $f ( x ) = \sqrt { 3 x ^ { 2 } + 5 x }$ at the point where x = 1 is $y = 2 x - 1$ .

Find the equation of the tangent line to the curve $f ( x ) = x ^ { 3 } - x ^ { 2 } + 1$ at the point (1, 1).

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) =5x³ - x² + 5

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$ .

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 the equation of the tangent line to the graph of f(x)=x2+1f ( x ) = x ^ { 2 } + 1f(x)=x2+1 at the point (4, 17).

Find dydx\frac { d y } { d x }dxdy​ if y=u3+7u2−6y = u ^ { 3 } + 7 u ^ { 2 } - 6y=u3+7u2−6 and u=x2+x−8u = x ^ { 2 } + x - 8u=x2+x−8 .

The equation of the line tangent to the graph of f(x)=x2+8xf ( x ) = x ^ { 2 } + 8 xf(x)=x2+8x at x = 7 is

What is the rate of change of f(t)=7t−5t+2f ( t ) = \frac { 7 t - 5 } { t + 2 }f(t)=t+27t−5​ with respect to t when t = 2?

Find dydx\frac { d y } { d x }dxdy​ , where x+y=xy\sqrt { x } + \sqrt { y } = x yx​+y​=xy .

If the position of an object moving along a straight line is given by s(t)=t3−7t2+8ts ( t ) = t ^ { 3 } - 7 t ^ { 2 } + 8 ts(t)=t3−7t2+8t at time t, find the object's velocity as a function of time.

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

If f(x)=x2−xf ( x ) = x \sqrt { 2 - x }f(x)=x2−x​ , then f′′(x)=0f ^ { \prime \prime } ( x ) = 0f′′(x)=0 at x = 0 and x = 2.

Find dydx\frac { d y } { d x }dxdy​ , where 3x+12y=5\frac { 3 } { x } + \frac { 1 } { 2 y } = 5x3​+2y1​=5 .

The equation of the line tangent to the graph of f(x)=x2+2xf ( x ) = x ^ { 2 } + 2 xf(x)=x2+2x at x = 7 is

The displacement function of a moving object is described by s(t)=t3+5t−4s ( t ) = t ^ { 3 } + 5 t - 4s(t)=t3+5t−4 . What is the velocity of the object as a function of time?

It is estimated that t years from now, the population of a certain suburban community will be p(t)=20−24t+3p ( t ) = 20 - \frac { 2 } { 4 t + 3 }p(t)=20−4t+32​ thousand people. At what rate will the population be growing 3 years from now?

If f(x)=3x−1x+1f ( x ) = \frac { 3 x - 1 } { x + 1 }f(x)=x+13x−1​ , what is f′(x)f ^ { \prime } ( x )f′(x) ?

Find an equation for the tangent line to the curve y=2+15xy = \sqrt { 2 + \frac { 1 } { 5 } x }y=2+51​x​ at the point where x = -1.

Differentiate: f(x)=x+7xf ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }f(x)=x​+x​7​

An equation for the tangent line to the curve f(x)=3x2+5xf ( x ) = \sqrt { 3 x ^ { 2 } + 5 x }f(x)=3x2+5x​ at the point where x = 1 is y=2x−1y = 2 x - 1y=2x−1 .

Find the equation of the tangent line to the curve f(x)=x3−x2+1f ( x ) = x ^ { 3 } - x ^ { 2 } + 1f(x)=x3−x2+1 at the point (1, 1).

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) =5x3 - x2 + 5

If x2y+xy2=7x ^ { 2 } y + x y ^ { 2 } = 7x2y+xy2=7 , then dydx=2xy+y2\frac { d y } { d x } = 2 x y + y ^ { 2 }dxdy​=2xy+y2 .

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 the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (4, 17).

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 7 u ^ { 2 } - 6$ and $u = x ^ { 2 } + x - 8$ .

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 8 x$ at x = 7 is

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

If the position of an object moving along a straight line is given by $s ( t ) = t ^ { 3 } - 7 t ^ { 2 } + 8 t$ at time t, find the object's velocity as a function of time.

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

Find $\frac { d y } { d x }$ , where $\frac { 3 } { x } + \frac { 1 } { 2 y } = 5$ .

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 is

The displacement function of a moving object is described by $s ( t ) = t ^ { 3 } + 5 t - 4$ . What is the velocity of the object as a function of time?

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 20 - \frac { 2 } { 4 t + 3 }$ thousand people. At what rate will the population be growing 3 years from now?

If $f ( x ) = \frac { 3 x - 1 } { x + 1 }$ , what is $f ^ { \prime } ( x )$ ?

Find an equation for the tangent line to the curve $y = \sqrt { 2 + \frac { 1 } { 5 } x }$ at the point where x = -1.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

An equation for the tangent line to the curve $f ( x ) = \sqrt { 3 x ^ { 2 } + 5 x }$ at the point where x = 1 is $y = 2 x - 1$ .

Find the equation of the tangent line to the curve $f ( x ) = x ^ { 3 } - x ^ { 2 } + 1$ at the point (1, 1).

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) =5x³ - x² + 5

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$ .