Deck 3: Differentiation Rules

Question 1

Find the linear approximation to $f ( x ) = e ^ { - 3 x }$ at a = 0.

A)1 + x
B)1 + 3x
C)1 - x
D)1 - 3x
E)3 + 3x
F)-3x
G)3 - 3x
H)None of these

Answer 1

1 - 3x

Question 2

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 25.05 }$ .

A)5.001
B)5.002
C)5.003
D)5.004
E)5.005
F)5.006
G)5.007
H)5.008

Answer 2

5.005

Question 3

Let (a) Find a linear approximation of f at x = 8:
(b) Use this linear approximation to estimate the value of the function at 7, 9, 7.99, and 8.01.(c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval [7; 9]. What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Answer 3

(a) (b), (c) The linear approximation is a good approximation to when x is near 8.
(d)

Question 4

Let $y = x ^ { 4 } + x ^ { 2 } + 1$ , $x = 1$ ,and dx = 1. Find the value of the differential dy.

A)2
B)4
C)6
D)8
E)10
F)0.12
G)0
H) $\frac { 1 } { 2 }$

Question 5

(a) Find the linearization of the function f (x) = sin x when x = 0.(b) Use these results to approximate sin (0.05) and sin (-0.005).

Question 6

Let y = $x ^ { 2 }$ , x = 2, and $\Delta$
x = 1. Find the value of the differential dy.

A)2

B) $\frac { 1 } { 2 }$

C) $\frac { 1 } { 3 }$

D)4

E) $\frac { 1 } { 4 }$

F) $\frac { 1 } { 8 }$

G)3

H)1

Question 7

Find the linear approximation of the function f (x) = $\sqrt { x + 24 }$ at x $1$ = 1 and use it to approximate $\sqrt { 24.98 }$ .

A)4.888
B)4.948
C)4.958
D)4.968
E)4.978
F)4.988
G)4.998
H)4.9995

Question 8

Use differentials to approximate $\sqrt { 16.2 }$ .

A)4.026
B)4.03
C)4.025
D)4.05
E)4.015
F)0.02498
G)4.0185
H)4.0245

Question 9

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated volume.

Question 10

During the flood of 1997 in Fargo, North Dakota, the Red River of the North rose menacingly during the month of April. Suppose that the official river level at noon on day t is given by the function R(t). We know that R(April 9) = 35.5 feet and $R ^ { \prime }$ (April 9) = 1.
(a) What are the units of $R ^ { \prime }$ (t)?
(b) Construct a linear function to estimate the water level of the Red River for dates near April 9.
(b) to estimate R(April 7) and R(April 11).
(c) Use this model from part
(d) The Red River crested on April 17 at 39.79 feet. What is $R ^ { \prime }$ (April 17)?
(e) If the Red River level on April 22, R(April 22), was 39 feet and $R ^ { \prime }$ (April 22) was -0.3, use a linear model to estimate R(April 26).

Question 11

Let $y = x ^ { 2 }$ , x = 3, and $\Delta$ x = 1. Find the value of the corresponding change $\Delta$ y.

A)4
B)8
C)7
D)2
E)3
F)6
G)1
H)5

Question 12

Show that for sufficiently small values of h,

Question 13

Find the linear approximation to $f ( x ) = \frac { 1 } { \sqrt { x ^ { 3 } + 1 } }$ at $a = 2 .$

A)-2x + 7
B) $\frac { 1 } { 3 } x - \frac { 2 } { 3 }$
C) $- \frac { 2 } { 9 } x + \frac { 7 } { 9 }$
D) $\frac { 1 } { 3 } x - \frac { 7 } { 3 }$
E) $2 x - 7$
F)
$- \frac { 1 } { 3 } x + \frac { 2 } { 3 }$
G)
$\frac { 2 } { 9 } x - \frac { 7 } { 9 }$
H) $- \frac { 1 } { 3 } x + \frac { 7 } { 3 }$

Question 14

Use differentials to approximate $\sqrt { 26 }$ .

A)5.1
B)5.2
C)5.15
D)5.3
E)5.4
F)5.25
G)5.35
H)5.05

Question 15

The period of a pendulum is given by the formula T = 2 $\pi$ $\sqrt { \frac { L } { g } }$
, where L is the length of the pendulum in feet, g = 32 ft/ $S ^ { 2 }$ is the acceleration due to gravity, and T is the length of one period in seconds. If the length of the pendulum is measured to be three feet long to within $\pm \frac { 1 } { 8 }$ inch, what is the approximate percentage error in the calculated period, T?

Question 16

Find the linear approximation to $f ( x ) = \frac { 1 } { ( 2 + x ) ^ { 3 } }$ at $a = 0$

A)
$\frac { 1 } { 8 } - \frac { 1 } { 8 } x$
B)
$\frac { 1 } { 8 } + \frac { 1 } { 8 } x$
C)
$\frac { 1 } { 8 } + \frac { 3 } { 16 } x$
D)
$\frac { 1 } { 8 } + \frac { 1 } { 16 } x$
E) $\frac { 1 } { 8 } - \frac { 3 } { 16 } x$
F) $\frac { 1 } { 8 } - \frac { 3 } { 4 } x$
G) $\frac { 1 } { 8 } - \frac { 1 } { 16 } x$
H)
$\frac { 1 } { 8 } + \frac { 3 } { 4 } x$

Question 17

The linear approximation of a function is useful only if the change in x is small. Illustrate this fact by approximating by regarding 18 to be "near" 36 instead of 16.

Question 18

Use differentials to approximate the change in the function $f ( x ) = x ^ { 2 } + x - 2$ when x varies from 1 to 1.01.

Question 19

A side of a square field is measured to be 144 feet with a possible error of 1 inch.(a) Use differentials to estimate the maximum error in the calculated area of the field.(b) What is the relative error?

Question 20

The diameter of a sphere is measured to be 6 inches with a possible error of 0.05 inches. Use differentials to estimate the maximum error in the calculated surface area.

Question 21

A particle moves along a straight line with equation of motion $s = t ^ { 3 } - t ^ { 2 }$ . Find the value of t at which the acceleration is equal to zero.

A) $- \frac { 2 } { 3 }$
B) $- \frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 3 }$
E) $- \frac { 1 } { 2 }$
F) $\frac { 1 } { 2 }$
G) $- \frac { 3 } { 2 }$
H)
$\frac { 3 } { 2 }$

Question 22

A particle moves along a straight line with equation of motion $S = t ^ { 3 } = 2 t$ . Find the smallest value of its velocity (for t ≥ 0).

A) $\frac { 1 } { 2 }$
B)-2
C) $\frac { 1 } { 3 }$
D) $- \frac { 1 } { 2 }$
E)-3
F)2
G)3
H) $- \frac { 1 } { 3 }$

Question 23

A stone is thrown into a pond, creating a circular wave whose radius increases at the rate of 1 foot per second. In square feet per second, how fast is the area of the circular ripple increasing 3 seconds after the stone hits the water?

A) $\pi$
B)2 $\pi$
C) $\frac { \pi } { 3 }$
D)6 $\pi$
E)3 $\pi$
F) $\frac { \pi } { 6 }$
G) $\frac { \pi } { 12 }$
H) $\frac { \pi } { 2 }$

Question 24

Profit (in dollars) for a company when x units of a certain product are produced is given by when x > 1.(a) What is the marginal profit (the derivative of the profit function)?
(b) If the current production level is x = 15, is the profit increasing or decreasing?
(c) If the current production level is x = 40, is the profit increasing or decreasing?
(d) At approximately what production level does the profit function reach its maximum value? What is the maximum profit?

Question 25

The cost function of manufacturing x meters of a fabric is .

Question 26

Suppose that a baseball is tossed straight upward and that its height (in feet) as a function of time (in seconds) is given by the formula h(t) = 128t - 16t².
(a) Find the instantaneous velocity and acceleration of the baseball at time t.
(b) What is the maximum height attained by the ball?
(c) What is the average velocity of the ball during the time interval from t = 1 to t = 4?
(d) How long does it take before the ball lands?
(e) At what time is the height of the ball 112 feet?

Question 27

Show that the rate of change of the circumference of a circle, with respect to the radius of the circle, does not depend on the radius.

Question 28

Let (a) Find a linear approximation of f at x = 1.(b) Use this linear approximation to predict the value of the function at -1, 0, 0.9, 1.1, 2, and 3.(c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval [-1; 3]. What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Question 29

The population of a bacteria colony after t hours is given by P(t) = $2000 e ^ { 0.087 t }$ . Find the growth rate of the colony when t = 16 hours.

A)0.087
B)16
C)174
D)700
E)1600
F)2000
G)20,000
H)32,000

Question 30

The mass of a rod varies in such a way that the mass of a piece x meters long, measured from the left end, is $x ^ { 2 }$ kilograms. Find the density in kg/m at the point 2 meters from the left end.

A)1
B)5
C)3
D)4
E)7
F)6
G)2
H)8

Question 31

If the total cost for producing x units of a particular product is given by C(x), then the average cost of production those x units is given by (a) If our cost function is , what value of x will result in the minimum average cost?
(b) What is the minimum average cost?

Question 32

A physics experiment involving the acceleration of shuttle on a rail produced the following
data: (a) Make a scatter plot of the data.(b) Fit a quadratic model to the data.(c) Based on your model, what was the initial position of the shuttle?
(d) Using your model, estimate the velocity of the shuttle when t = 0.1 and when t = 0.45 seconds.(e) What is your estimated acceleration of the shuttle when t = 0.1 and when t = 0.45 seconds?

Question 33

Let f (x) = .(a) Find a linear approximation of f at x = (b) Use this linear approximation to predict the value of the function at and (c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval What does the graph tell you about the size of the difference between the function values and the linear approximation values?

Question 34

The cost function of manufacturing x meters of a fabric is C (x) = 25,000 + 3x - $0.002 x ^ { 2 }$ + $0.000001 x ^ { 3 }$ . Find $\mathrm { C } ^ { ' }$ (5000).

A)58
B)580
C)5.8
D)5800
E)60
F)600
G)6
H)6000

Question 35

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 3 t + 2$ . Find the value of t at which the particle reverses its direction.

A) $\frac { 1 } { 2 }$
B)0
C) $\frac { 3 } { 2 }$
D) $\frac { 2 } { 3 }$
E)1
F)2
G) $\frac { 3 } { 4 }$
H)
$\frac { 4 } { 3 }$

Question 36

The position of a particle is given by the function s(t) = , where t is measured in seconds and s in meters.(a) Find the velocity at time t.(b) When is the particle at rest?
(c) When is the particle moving in the positive direction?
(d) Draw a diagram to represent the motion of the particle.(e) Find the total distance traveled by the particle during the time interval [1,3].

Question 37

A particle moves along a straight line with equation of motion $s = t ^ { 2 } - 2 t$ . Find the instantaneous velocity of the particle at time t = 1.

A)1
B)4
C)0
D)3
E)8
F)6
G)5
H)2

Question 38

The relationship between the rate of a certain chemical reaction and temperature under certain circumstances is given by $R ( T ) = 0.1 \left( - 0.05 T ^ { 3 } + 4 T ^ { 2 } + 120 \right)$ grams/sec, where R is the rate of reaction and T is the temperature (in žC).
(a) Find the temperature T at which the reaction rate reaches its maximum.
(b) What is the maximum reaction rate?

Question 39

Suppose the amount of a drug left in the body t hours after administration is mg. In mg=h, find the rate of decrease of the drug 4 hours after administration.

Question 40

Below is a table of the vapor pressure (in kilopascals) of water for various temperatures (in degrees Kelvin): (a) Estimate the rate of change of pressure with respect to temperature on the following intervals:
(i) [363, 373]
(ii) [333, 343]
(iii) [273, 283]
(b) Plot the points from the table and fit an appropriate exponential model to these data.(c) From the model in part (b), determine the instantaneous rate of change of pressure with respect to temperature.(d) Is the rate of change of pressure increasing or decreasing with respect to temperature? Justify your answer.

Question 41

Let $f ( x ) = \log _ { 3 } x ^ { 2 }$ . Find the value of $f ^ { \prime } ( 2 )$ .

A) $3 ^ { e }$
B)ln 3
C)ln $\left( \frac { 1 } { 3 } \right)$
D)1
E) $\frac { 1 } { \sqrt { 3 } }$
F) $\frac { 1 } { \ln 3 }$
G)
$\frac { 1 } { 3 }$
H) $\sqrt { 3 }$

Question 42

Let $f ( x ) = ( \sqrt { x } ) ^ { x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

A)0
B) $\frac { 1 } { 4 }$ f.2
C) $\frac { 1 } { 2 }$ g.4
D) $\frac { 3 } { 4 }$ h.6

E)1

Question 43

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is concave upward.

A) $( 0 , \infty )$
B) $( 0,1 ]$
C) $[ 1 , \infty )$
D) $\left( 0 , e ^ { - 1 } \right]$
E) $\left[ e ^ { - 1 } , \infty \right)$
F) $\left( 0 , e ^ { - 2 } \right]$
G) $\left[ e ^ { - 2 } , \infty \right)$
H) $( 0 , \sqrt { e } )$

Question 44

Let $f ( x ) = x \ln \left( x ^ { 2 } - 3 \right)$ . Find the value of $f ^ { \prime } ( 2 )$ .

A)0
B)2
C)4
D)6
E)8
F)10
G)12
H)14

Question 45

Let $f ( x ) = \log _ { 2 } x$ . Find the value of $f ^ { \prime } ( 1 )$ .

A)2
B) $e ^ { 2 }$
C)ln $\left( \frac { 1 } { 2 } \right)$
D) $e ^ { - \frac { 1 } { 2 } }$
E) $2 ^ { e }$
F) $\frac { 1 } { 2 }$
G) $\frac { 1 } { \ln 2 }$
H) $e ^ { \frac { 1 } { 2 } }$

Question 46

The following table shows the relationship between pressure (in atmospheres) and volume (in liters) of hydrogen gas at 0 °C. (a) Find the average rate of change of volume with respect to pressure for the following pressure intervals:
(i) [1, 3]
(ii) [2, 3]
(iii) [4, 5]
(b) Plot the data points and fit an appropriate power function to these data.(c) Use the model from part (b) and determine the instantaneous rate of change of volume with respect to pressure.(d) Compare the instantaneous rate at P = 5 with the average rate for [4, 5]. Which is larger? Why is this the case?

Question 47

Find the interval on which the graph of $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$ is concave upward.

A)(-1, 1)
B)(-1, 2)
C)(-2, 1)
D)(-2, 2)
E)(-1, 3)
F)(-3, 2)
G)(-3, 3)
H) $( - \infty , \infty )$

Question 48

Let $f ( x ) = x ^ { 2 x }$ . Find the value of $f ^ { \prime } ( 1 )$ .

A)2
B)e - 1
C) $e ^ { e } - 1$
D)e + 1
E)e
F) $e ^ { e + 1 }$
G) $e ^ { e - 1 }$
H) $e ^ { 2 }$

Question 49

Find an equation of the tangent line to the graph of at the point .

Question 50

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is increasing.

A) $( 0 , \infty )$
B) $( 0,1 ]$
C) $[ 1 , \infty )$
D) $\left( 0 , e ^ { - 1 } \right]$
E) $\left[ e ^ { - 1 } , \infty \right)$
F) $\left( 0 , e ^ { - 2 } \right]$
G) $\left[ e ^ { - 2 } , \infty \right)$
H) $( 0 , \sqrt { e } )$

Question 51

Let $f ( x ) = x ^ { \sqrt { x } }$ . Find the value of $f ^ { \prime } ( 4 )$ .

A)2
B)4
C)8
D)16
E)2 + ln 4
F)4 + ln 2
G)8 + 4ln 4
H)16 + 4ln 2

Question 52

Find an equation of the tangent line to the graph of at the point (1, 0).

Question 53

Let $f ( x ) = x ^ { \frac { 1 } { x } }$ . Find the value of $f ^ { \prime } ( e )$ .

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 54

Let $f ( x ) = \log _ { 10 } \pi ^ { x }$ . Find the value of $f ^ { \prime } ( 100 )$ .

A) $\pi$
B)10 $\pi$
C) $\pi$ ¹⁰
D)100 $\pi$
E)
$\pi$ ¹⁰⁰
F)
log₁₀
$\pi$
G)10 log₁₀
$\pi$
H)100 log₁₀
$\pi$

Question 55

Find the minimum value of $f ( x ) = x ^ { - \frac { 1 } { x } }$ .

A)0
B) $e ^ { - \frac { 1 } { \ell } }$
C) $e ^ { \frac { 1 } { e } }$
D)e
E) $e ^ { e }$
F) $e ^ { -e }$
G)1
H)Does not exist

Question 56

Let $f ( x ) = \ln ( \ln ( x ) )$ . Find the value of $f ^ { \prime } ( e )$ .

A) $\frac { 1 } { e ^ { 2 } }$
B) $e ^ { 2 }$
C)1 + e
D) $\frac { 1 } { e }$
E)ln 2
F)1
G)e
H)0

Question 57

The following table shows the concentration (in mol/L) of a certain chemical in terms of reaction time (in hours) during a decomposition reaction. (a) Find the average rate of change of concentration with respect to time for the following time intervals:
(i) [0, 5]
(ii) [10, 20]
(iii) [30, 50]
(b) Plot the points from the table and fit an appropriate exponential model to the data.(c) From your model in part (b), determine the instantaneous rate of change of concentration with respect to time.(d) Is the rate of change of concentration increasing or decreasing with respect to time? Justify your answer.

Question 58

Let $f ( x ) = ( \sin x ) ^ { x }$ . Find the value of $f ^ { \prime } \left( \frac { \pi } { 2 } \right)$ .

A)0
B)1
C)2
D)
$e ^ { \frac { \pi } { 2 } }$
E) $e ^ { x }$
F) $\frac { \pi } { 2 }$
G) $\pi$
H)2 $\pi$

Question 59

Let $f ( x ) = \ln \left( \sin ^ { 2 } x + 1 \right)$ . Find the value of $f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

A)0
B) $\frac { 1 } { 2 }$
C) $\frac { 2 } { 3 }$
D)1
E) $\frac { 3 } { 2 }$
F)
$\sqrt { 2 }$
G)2
H)3

Question 60

Find the x-coordinate of the point at which the graph of has a horizontal tangent.

Question 61

Find the exact value of $\sin ^ { - 1 } \left( \sin \frac { 5 \pi } { 6 } \right)$ .

A) $- \frac { 5 \pi } { 6 }$
B) $- \frac { 2 \pi } { 3 }$
C) $- \frac { \pi } { 6 }$
D)0
E) $\frac { \pi } { 6 }$
F) $\frac { \pi } { 3 }$
G) $\frac { 2 \pi } { 3 }$
H) $\frac { 5 \pi } { 6 }$

Question 62

Differentiate the following functions:
(a) (b) (c)

Question 63

Find the x-coordinate of the point at which the graph of has a horizontal tangent.

Question 64

Let $f ( x ) = \tan ^ { - 1 } x$ . Find the value of $f ^ { \prime } ( 1 )$ .

A) $- \frac { 1 } { 2 }$
B) $\frac { 1 } { 2 }$
C) $- \frac { 1 } { 3 }$
D)1
E) $\frac { 1 } { 3 }$
F) $- \frac { 1 } { 4 }$
G) $\frac { 1 } { 4 }$
H) $- 1$

Question 65

Let $f ( x ) = \sin ^ { - 1 } \left( \frac { 4 } { x } \right)$ . Find the value of $f ^ { \prime } ( 8 )$ .

A) $- \frac { 1 } { 8 \sqrt { 3 } }$
B) $- \frac { 1 } { 4 \sqrt { 3 } }$
C) $- \frac { 1 } { 2 \sqrt { 3 } }$
D) $- \frac { 1 } { \sqrt { 3 } }$
E) $\frac { \pi } { 8 }$
F) $\frac { \pi } { 4 }$
G) $\frac { \pi } { 2 }$
H) $\pi$

Question 66

Find .(a) (b) (c)

Question 67

Simplify the expression .

Question 68

Find the critical numbers for and identify each as a relative maximum, relative minimum, or neither.

Question 69

Let $f ( x ) = \tan ^ { - 1 } x + \tan ^ { - 1 } \frac { 1 } { x }$ .(a) Show that $f ( x )$ is constant on $( - \infty , 0 ) \text { and } ( 0 , \infty )$ .(b) Determine the value of the constant(s).

Question 70

Find the domain of the function $f ( x ) = \sin ^ { - 1 } ( 3 - 2 x )$ .

A) $[ 0,1 ]$
B) $[ 1,2 ]$
C) $[ 0,2 ]$
D) $[ 0,3 ]$
E) $[ 1,3 ]$
F) $[ 2,3 ]$
G) $[ - 3 , - 1 ]$
H) $[ - 3 , - 2 ]$

Question 71

Let $f ( x ) = \tan ^ { - 1 } \left( x ^ { 2 } + 1 \right)$ . Find the value of $f ^ { \prime } ( 1 )$ .

A)0
B)0.1
C)0.2
D)0.3
E)0.4
F)0.5
G)0.6
H)0.8

Question 72

Find the derivatives of the following functions:
(a) (b) (c) (d) (e)

Question 73

If $x ^ { 2 } + y ^ { 2 } = 25$ find the value of $\frac { d y } { d x }$ at the point (3, 4).

A) $\frac { 3 } { 5 }$
B) $- \frac { 3 } { 5 }$
C) $- \frac { 4 } { 5 }$
D) $\frac { 3 } { 4 }$
E)0
F)1
G) $\frac { 4 } { 5 }$
H) $- \frac { 3 } { 4 }$

Question 74

Let $f ( x ) = \sin ^ { - 1 } ( 2 x )$ . Find the value of $f ^ { \prime } ( 0 )$ .

A) $- \frac { 1 } { 2 }$
B).2
C) $- 2$
D)1
E) $\frac { 1 } { 2 }$
F) $- 1$
G).0
H) $- \frac { 1 } { \sqrt { 2 } }$

Question 75

Simplify the express: $\sec \left( \tan ^ { - 1 } x \right)$ .

A)1
B) $x ^ { 2 }$
C) $\sqrt { x }$
D) $\sqrt { x ^ { 2 } + 1 }$
E) $\frac { 1 } { x }$
F) $\frac { 1 } { \sqrt { x ^ { 2 } + 1 } }$
G) $\frac { x } { \sqrt { x ^ { 2 } + 1 } }$
H)
$\frac { \sqrt { x ^ { 2 } + 1 } } { x }$

Question 76

Find the exact value of $\tan ^ { - 1 } ( \cos \pi )$ .

A) $- \frac { \pi } { 2 }$
B) $- \frac { \pi } { 4 }$
C)0
D) $\frac { \pi } { 4 }$
E) $\frac { 3 \pi } { 4 }$
F) $\frac { 5 \pi } { 6 }$
G) $\frac { \pi } { 2 }$
H) $\pi$

Question 77

Find .(a) (b)

Question 78

Find the exact value of $\tan \left( \cos ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)$ .

A)1
B) $\sqrt { 3 }$
C) $\sqrt { 2 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E)0
F) $\frac { \pi } { 4 }$
G) $\frac { \pi } { 3 }$
H) $\frac { \pi } { 6 }$

Question 79

Differentiate the following functions:
(a) (b) (c)

Question 80

Find implicitly:
(a) (b)