Question 1

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 7 x ^ { 4 } + 4 x ^ { 3 } - 6 x ^ { 2 } + 11 x - 7$ .

Accepted Answer

A)  $f ^ { ( 4 ) } ( x ) = 60 x ^ { 2 } - 168 x + 24$ 
B)  $f ^ { ( 4 ) } ( x ) = 120 x - 168$ 
C)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 7 x$ 
D)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 7 x + 4$

Question 2

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (2, 5).

Accepted Answer

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

Question 3

If the displacement of a moving object is $s ( t ) = t ^ { 3 }$ , the acceleration is 6t.

Accepted Answer

A) True 
 B)False

Question 4

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f ( x ) = - x ^ { 3 } + 10 x ^ { 2 } - 3 x$ transistor radios x hours later. Approximately how many radios will the worker assemble between 10:00 and 10:15 A.M.?

Accepted Answer

A)  Approximately 25 radios 
B)  Approximately 375 radios 
C)  Approximately 6 radios 
D)  Approximately 26 radios

Question 5

If $g ( y ) = \sqrt { 5 y + y ^ { 2 } }$ represents the height in inches of a sapling y weeks after germination, find $g ^ { \prime } ( 6 )$ and interpret what it tells us about the height of the tree. Round your answer to 1 decimal place.

Accepted Answer

The answer of If \(g ( y ) = \sqrt...

Question 6

When toasters are sold for p dollars apiece, local consumers will buy $D ( p ) = \frac { 57,600 } { p }$ toasters a month. It is estimated that t months from now, the price of the toasters will be $p ( t ) = 0.03 t ^ { 3 / 2 } + 22.08$ dollars. Compute the rate at which the monthly demand for the toasters will be changing with respect to time 16 months from now.

Accepted Answer

The answer of When toasters are sold for p dollars...

Question 7

It is estimated that t years from now, the population of a certain suburban community will be $p ( t ) = 30 - \frac { 7 } { 2 t + 1 }$ thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be $C ( p ) = 0.3 \sqrt { p ^ { 2 } + p + 30 }$ parts per million (ppm) when the population is p thousand. The rate at which the level of pollution is changing with respect to time 3 years from now is about 0.084 ppm per year.

Accepted Answer

A) True 
 B)False

Question 8

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

Accepted Answer

A) True 
 B)False

Question 9

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...

Question 10

An equation for the tangent line to the curve $f ( x ) = x ^ { 3 } ( 1 - 3 x ) ^ { 2 }$ at the point where x = -1 is y = 72x + 56.

Accepted Answer

A) True 
 B)False

Question 11

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

Question 12

When a certain commodity is sold for p dollars per unit, consumers will buy $D ( p ) = \frac { 30,000 } { p }$ units per month. It is estimated that t months from now, the price of the commodity will be $p ( t ) = 0.3 t ^ { 5 / 2 } + 5.4$ dollars per unit. The monthly demand will be decreasing 40 months from now.

Accepted Answer

A) True 
 B)False

Question 13

If $f ( x ) = \sqrt [ 3 ] { x } - \frac { 1 } { \sqrt { x } }$ , differentiate f (x).

Accepted Answer

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

Question 14

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at $\left( 2 , \frac { 1 } { 2 } \right)$ .

Accepted Answer

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

Question 15

Differentiate: $f ( x ) = \left( x ^ { 2 } + 2 \right) ( x + 4 )$

Accepted Answer

A)  $3 x ^ { 2 } + 8 x + 2$ 
B)  2x + 1 
C)  16x + 1 
D)  $x ^ { 2 } + 1$

Question 16

Find the equation of the tangent line to the given curve at the specified point: $x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8$ ; (0, 8)

Accepted Answer

A)  $y = \frac { 1 } { 35 } x + 8$ 
B)  $y = - \frac { 1 } { 35 } x + 8$ 
C)  y = -35x + 8 
D)  y = 35x + 8

Question 17

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1.f (x) = x³ + 5x + 2

Accepted Answer

The answer of Find the rate of change of the...

Question 18

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$

Accepted Answer

A)  $\frac { 6 x - y ^ { 3 } } { 3 x y ^ { 2 } - 7 }$ 
B)  $y ^ { 3 } - 6 x$ 
C)  $\frac { 6 x ^ { 2 } } { y ^ { 3 } }$ 
D)  $y ^ { 3 } - 6 x - 7$

Question 19

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x + 9$ at the point (1, 12).

Accepted Answer

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

Question 20

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$ .

Accepted Answer

A) True 
 B)False

Exam 2: Differentiation: Basic Concepts

Find f(4) (x) f ^ { ( 4 ) } ( x ) f(4) (x) if f(x) =x5−7x4+4x3−6x2+11x−7f ( x ) = x ^ { 5 } - 7 x ^ { 4 } + 4 x ^ { 3 } - 6 x ^ { 2 } + 11 x - 7f(x) =x5−7x4+4x3−6x2+11x−7 .

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 (2, 5).

If the displacement of a moving object is s(t)=t3s ( t ) = t ^ { 3 }s(t)=t3 , the acceleration is 6t.

If g(y)=5y+y2g ( y ) = \sqrt { 5 y + y ^ { 2 } }g(y)=5y+y2​ represents the height in inches of a sapling y weeks after germination, find g′(6)g ^ { \prime } ( 6 )g′(6) and interpret what it tells us about the height of the tree. Round your answer to 1 decimal place.

The function f(x)=x2x+1−5f ( x ) = \frac { x } { 2 x + 1 } - 5f(x)=2x+1x​−5 will decrease by approximately 0.6 as x decreases from 3 to 2.7.

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?

An equation for the tangent line to the curve f(x)=x3(1−3x)2f ( x ) = x ^ { 3 } ( 1 - 3 x ) ^ { 2 }f(x)=x3(1−3x)2 at the point where x = -1 is y = 72x + 56.

If f(x)=x3−1xf ( x ) = \sqrt [ 3 ] { x } - \frac { 1 } { \sqrt { x } }f(x)=3x​−x​1​ , differentiate f (x).

Find the equation of the tangent line to the graph of f(x) =1xf ( x ) = \frac { 1 } { x }f(x) =x1​ at (2,12) \left( 2 , \frac { 1 } { 2 } \right) (2,21​) .

Differentiate: f(x) =(x2+2) (x+4) f ( x ) = \left( x ^ { 2 } + 2 \right) ( x + 4 ) f(x) =(x2+2) (x+4)

Find the equation of the tangent line to the given curve at the specified point: x2y5−4xy=3x+y−8x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8x2y5−4xy=3x+y−8 ; (0, 8)

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1.f (x) = x3 + 5x + 2

Find dydx\frac { d y } { d x }dxdy​ , where xy3−3x2=7yx y ^ { 3 } - 3 x ^ { 2 } = 7 yxy3−3x2=7y

Find the equation of the tangent to the graph of f(x)=x2+2x+9f ( x ) = x ^ { 2 } + 2 x + 9f(x)=x2+2x+9 at the point (1, 12).

If x2+3xy+y2=15x ^ { 2 } + 3 x y + y ^ { 2 } = 15x2+3xy+y2=15 , then dydx=2x+3y\frac { d y } { d x } = 2 x + 3 ydxdy​=2x+3y .

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 7 x ^ { 4 } + 4 x ^ { 3 } - 6 x ^ { 2 } + 11 x - 7$ .

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at the point (2, 5).

If the displacement of a moving object is $s ( t ) = t ^ { 3 }$ , the acceleration is 6t.

If $g ( y ) = \sqrt { 5 y + y ^ { 2 } }$ represents the height in inches of a sapling y weeks after germination, find $g ^ { \prime } ( 6 )$ and interpret what it tells us about the height of the tree. Round your answer to 1 decimal place.

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

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?

An equation for the tangent line to the curve $f ( x ) = x ^ { 3 } ( 1 - 3 x ) ^ { 2 }$ at the point where x = -1 is y = 72x + 56.

If $f ( x ) = \sqrt [ 3 ] { x } - \frac { 1 } { \sqrt { x } }$ , differentiate f (x).

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 1 } { x }$ at $\left( 2 , \frac { 1 } { 2 } \right)$ .

Differentiate: $f ( x ) = \left( x ^ { 2 } + 2 \right) ( x + 4 )$

Find the equation of the tangent line to the given curve at the specified point: $x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8$ ; (0, 8)

Find the rate of change of the given function f (x) with respect for x for the prescribed value x = -1.f (x) = x³ + 5x + 2

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x + 9$ at the point (1, 12).

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$ .