Deck 2: Differentiation: Basic Concepts

Question 1

You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters.If your measurement of the side is accurate to within 2%,approximately how accurate is your calculation of this volume? Round to two decimal places,if necessary.

A)Maximum error in volume is about ±164.64 cm³
B)Maximum error in volume is about ±0.84 cm³
C)Maximum error in volume is about ±2,304.96 cm³
D)Maximum error in volume is about ±11.76 cm³

Answer 1

Maximum error in volume is about ±164.64 cm³

Question 2

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4$ .

A) $f ^ { ( 4 ) } ( x ) = 120 x - 192$
B) $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x$
C) $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x + 7$
D) $f ^ { ( 4 ) } ( x ) = 60 x ^ { 2 } - 192 x + 42$

Answer 2

$f ^ { ( 4 ) } ( x ) = 120 x - 192$

Question 3

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

A)(0,0)and (-4,-8)
B)There are none.
C)(0,0)
D)(2,1)

Answer 3

(0,0)and (-4,-8)

Question 4

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ ,find the average rate of change of f (x)with respect to x as x changes from 144 to 145.Then use calculus to find the instantaneous rate of change at x = 144.Round your answer to six decimal places,if necessary.

A)Average rate of change: 0.000864; Instantaneous rate of change: 0.000868
B)Average rate of change: -0.000864; Instantaneous rate of change: 0.125
C)Average rate of change: 0.000864; Instantaneous rate of change: -0.125
D)Average rate of change: -0.000864; Instantaneous rate of change: 0.000868

Question 5

Find the rate of change of the given function f (x)with respect for x for the prescribed value x = -2. f (x)= x³ + 3x + 9

A)6
B)-3
C)15
D)24

Question 6

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

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 7

When a certain commodity is sold for p dollars per unit,consumers will buy $D ( p ) = \frac { 31,500 } { p }$ units per month.It is estimated that t months from now,the price of the commodity will be $p ( t ) = t ^ { 2 / 3 } + 5.15$ dollars per unit.The approximate rate at which the monthly demand will be changing with respect to time in 27 months is

A)-32 units per month
B)-35 units per month
C)-132 units per month
D)35 units per month

Question 8

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 3$ at the point (3,12).

A)y = 12
B)y = 6x - 6
C)x = 3
D)Not defined

Question 9

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 9 t ^ { 2 } + 15 t + 2$ .Find the velocity and acceleration of the object at time t.When is the object stationary?

A) $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 3
B) $v ( t ) = 3 t ^ { 2 } - 6 t + 15$ ; a(t)= 6t - 6; t = 1
C) $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1
D) $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1 and 5

Question 10

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

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

Question 11

An equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

A)y = 5x + 1
B)y = 5x - 1
C)y = 10x - 1
D)y = 10x + 1

Question 12

The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula $S = 4 \pi r ^ { 2 }$ to be no greater than 6 percent is about:

A)2%
B)6%
C)3%
D)1%

Question 13

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

Question 14

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

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

Question 15

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 6 }$

A)2x
B)-x
C) $\frac { 3 x ^ { 2 } + 12 x } { ( x - 6 ) ^ { 2 } }$
D) $\frac { x ^ { 2 } - 12 x } { ( x - 6 ) ^ { 2 } }$

Question 16

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 6 )$

A) $x ^ { 2 } + 1$
B) $3 x ^ { 2 } + 12 x + 1$
C)2x + 1
D)12x + 1

Question 17

Find $f ^ { \prime \prime \prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 3 } { x ^ { 2 } } + \sqrt { 7 }$

A) $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 16 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$
B) $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$
C) $f ^ { \prime \prime \prime } ( x ) = - \frac { 5 } { 72 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$
D) $f ^ { \prime \prime \prime } ( x ) = - \frac { 3 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 3 } { x ^ { 3 } }$

Question 18

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M.will have assembled $f ( x ) = - x ^ { 3 } + 7 x ^ { 2 } - 3 x$ transistor radios x hours later.Approximately how many radios will the worker assemble between 9:00 and 9:30 A.M.?

A)Approximately 7 radios
B)Approximately 13 radios
C)Approximately 14 radios
D)Approximately 390 radios

Question 19

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

Question 20

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

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 21

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)

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

Question 22

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

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 23

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

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

Question 24

Suppose the output at a certain factory is $$Q = 4 x ^ { 3 } + 5 x ^ { 3 } y ^ { 4 } + 3 y ^ { 3 }$$ units,where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor.The current labor force consists of 20 hours of skilled labor and 10 hours of unskilled labor.Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level.Round you answer to two decimal places,if necessary.

A)-2.67 hours
B)2.67 hours
C)-1 hours
D)-0.38 hours