Deck 3: Short-Cuts to Differentiation

Question 1

Given and , find .

Answer 1

Question 2

Consider the function .Is f increasing or decreasing at the point x = 0.5?

Answer 2

decreasing

Question 3

Find a formula for the slope of the tangent line to

A)2x -
B)2x - 9
C)2(x - 9)³
D)(x - 9)³/3
E)none of the above

Answer 3

2x -

Question 4

Find the derivative of .

A)
B)
C)
D)
E)None of the above

Question 5

Consider the function .Is f concave up or down at the point x = -0.2?

Question 6

If represents the position of a particle at time t seconds, then g'(t)represents the __________ of the particle at time t seconds.

Question 7

Find the derivative of .

A)
B)
C)
D)
E)None of the above

Question 8

The 12th derivative of is 0.

Question 9

If , then what is ?

Question 10

Given the variable , find when .

A)
B)
C)
D)

Question 11

A man plans to propose to a woman in romantic fashion by taking her up in an air balloon.Unfortunately, he pulls the diamond ring from his pocket and drops it over the side of the balloon's basket.The ring's position above the earth t seconds after it falls is given by the function feet.How fast is the ring falling 3 seconds after he drops it?

Question 12

A man plans to propose to a woman in romantic fashion by taking her up in an air balloon.Unfortunately, he pulls the diamond ring from his pocket and drops it over the side of the balloon's basket.The ring's position above the earth t seconds after it falls is given by the function feet.How fast is the ring falling at the instant it hits the ground? 1325

Question 13

Consider the function .Estimate using the tangent line at x = 1.

Question 14

Given , at which value(s)of x does the curve have a horizontal tangent?

A)1
B)2
C)3
D)4
E)5

Question 15

Find the derivative of .

A)
B)
C)
D)
E)None of the above

Question 16

Find when .

Question 17

Find when .

A)
B)
C)
D)all of the above
E)none of the above

Question 18

Given , what is the slope of the tangent line to the curve at x = -3?

Question 19

Find the derivative of with respect to x.

A)
B)
C)
D)6x - 3

Question 20

Differentiate .

A)
B)
C)
D)
E)None of the above

Question 21

Find the derivative of $y = 4 ^ { x } - 4$ .

A) $4 ^ { x }$
B) $( \ln 4 ) 4 ^ { x }$
C) $( \ln 4 ) 4 ^ { x } - \ln 4$
D) $x 4 ^ { x - 1 }$

Question 22

Find the derivative of $g ( x ) = 3 x - \frac { 1 } { \sqrt [ 3 ] { x } } + 3 ^ { x }$ .

A) $3 + \frac { 1 } { 3 x ^ { 4 / 3 } } + ( \ln 3 ) 3 ^ { x }$
B) $3 - \frac { 1 } { 3 x ^ { 4 / 3 } } + ( \ln 3 ) 3 ^ { x }$
C) $3 + \frac { 1 } { 3 \sqrt { x } } + x 3 ^ { x - 1 }$
D) $3 - \frac { 1 } { 3 \sqrt { x } } + x 3 ^ { x - 1 }$

Question 23

A child earns five cents from her grandfather for each dandelion she pulls out of his front yard.The child pulls out all the dandelions that are there.As the season passes, the number of dandelions in the front yard increase according to the model .After 15 days, her grandfather calls off the deal.How many dandelions does she pull on the 15th day? How fast is the number of dandelions increasing on the 15th day?

Question 24

Find the slope of the graph of at the point where it crosses the y-axis.

Question 25

Consider the following table of data for the function f.
$$\begin{array} { c c c c c c } x & 5.0 & 5.1 & 5.2 & 5.3 & 5.4 \\f ( x ) & 9.2 & 8.8 & 8.3 & 7.7 & 7.0\end{array}$$ Suppose g is a function such that g(5.1)= 9 and g'(5.1)= 3.Find h'(5.1)where h(x)= f(x)g(x).Use the right-hand estimate for $f ^ { \prime } ( 5.1 )$ .Round to 2 decimal places.

Question 26

Prove that the function , where a > 1 and b > 1, is increasing for all values of t.

Question 27

With a yearly inflation rate of 7%, prices are described by , where is the price in dollars when t = 0 and t is time in years.If = 1.3, how fast (in cents/year)are prices rising when t = 19? Round to 1 decimal place.

Question 28

On what intervals is the function :
a)increasing?
b)decreasing?
c)concave up?
d)concave down?

Question 29

Consider the following table of data for the function f. $$\begin{array} { c c c c c c } x & 5.0 & 5.1 & 5.2 & 5.3 & 5.4 \\f ( x ) & 9.1 & 8.7 & 8.2 & 7.6 & 6.9\end{array}$$ What is the sign of f '(5.1)?

A)positive
B)negative

Question 30

Find the derivative of $f ( x ) = ( \ln 4 ) x ^ { 2 } + \ln ( 7 ) e ^ { x }$ .

A) $2 \ln ( 4 x ) + 7 x$
B) $( \ln 8 ) x + \ln ( 7 ) e ^ { x }$
C) $( 2 \ln 4 ) x + 7 e ^ { x }$
D) $( 2 \ln 4 ) x + \ln ( 7 ) e ^ { x }$

Question 31

On what intervals is the polynomial concave down? Concave up?

Question 32

Given , find .

Question 33

Consider the graph $y = e ^ { x }$ .What is the x-intercept of the tangent line to the graph at $\left( a , e ^ { a } \right)$

A) $a - 1$
B)e ^a
$( 1 - a )$
C) $1 - a$
D) $e ^ { a } ( \alpha - 1 )$

Question 34

If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .If you solve this equation for P and hold F and r constant, what will the sign of $\frac { d P } { d t }$ be?

A)positive
B)negative

Question 35

Differentiate $g ( t ) = e ^ { - t } + 8 e ^ { - 8 t }$ .

A) $- e ^ { - t } - 64 e ^ { - 8 t }$
B) $e ^ { - t } + 64 e ^ { - 8 t }$
C) $t e ^ { - t - 1 } + 8 t e ^ { - 8 t - 1 }$
D) $- t e ^ { - t - 1 } - 64 t e ^ { - 8 t - 1 }$

Question 36

On what intervals is the polynomial increasing? Decreasing?

Question 37

Find the derivative of $h ( t ) = t ^ { \pi ^ { 4 } } + \left( \pi ^ { 4 } \right) ^ { t } + \pi t ^ { 4 }$ .

A) $\pi ^ { 4 } t ^ { \left( \pi ^ { 4 } - 1 \right) } + t \left( \pi ^ { 4 } \right) ^ { t - 1 } + 4 \pi t ^ { 3 }$
B) $\pi ^ { 4 } t ^ { \left( \pi ^ { 4 } - 1 \right) } + \left( \pi ^ { 4 } \right) ^ { + } \ln \left( \pi ^ { 4 } \right) + 4 \pi ^ { 3 }$
C) $t ^ { \left( \pi ^ { 4 } - 1 \right) } \ln t + \left( \pi ^ { 4 } \right) ^ { t } \ln \left( \pi ^ { 4 } \right) + 4 \pi t ^ { 4 } \ln t$
D) $4 t ^ { \pi ^ { 3 } } + 4 t \left( \pi ^ { 3 } \right) ^ { t - 1 } + 4 \pi t ^ { 4 } \ln t$

Question 38

If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .Assuming P and r are constant, find $\frac { d F } { d t }$ .

A) $P \left( 1 + \frac { r } { 100 } \right) ^ { t }$
B) $P t \left( 1 + \frac { r } { 100 } \right) ^ { t - 1 }$
C) $P \left( 1 + \frac { r } { 100 } \right) ^ { t } \ln \left( 1 + \frac { r } { 100 } \right)$
D) $\frac { P \left( 1 + \frac { r } { 100 } \right) ^ { t } } { \ln \left( 1 + \frac { r } { 100 } \right) }$

Question 39

Find the derivative of $g ( t ) = ( 1 / e ) ^ { t } + e ^ { t } + e$ .

A) $t ( 1 / e ) ^ { t - 1 } + t e ^ { t - 1 }$
B) $- t ( 1 / e ) ^ { t - 1 } + t e ^ { t - 1 }$
C) $- ( 1 / e ) ^ { t } + e ^ { t }$
D) $( 1 / e ) ^ { t } + e ^ { t }$

Question 40

Find the derivative of $f ( x ) = 7 e ^ { \pi x }$ .

A) $7 \pi x e ^ { \pi x - 1 }$
B) $7 \ln ( \pi ) e ^ { \pi x }$
C) $7 e ^ { \pi x }$
D) $7 \pi e ^ { \pi x }$

Question 41

Given $f ( x ) = \frac { x ^ { 3 } } { 2 x + 1 }$ , $g ( x ) = \frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } }$ , and h(x)= f(x)g(x), find h'(1).Round to 2 decimal places.

Question 42

Find the equation of the tangent line to at x = 3 and use it find the point where the tangent line crosses the x-axis.Round to 2 decimal places.

Question 43

The table below gives values for functions f and g, and their derivatives.
Find g(f(x))at x = -1.If is cannot be computed from the information given, enter "cannot find".

Question 44

Use the product rule to write a proof of the constant multiple rule: .

Question 45

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 0 )$ if $H ( x ) =$ e ^g(x)
$$+ \pi$$ x.
$\begin{array} { c c c c c } \boldsymbol { x } & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \\0 & 1 & - 1 & 2 & 5 \\1 & - 1 & 2 & 4 & 0 \\2 & 7 & 3 & 11 & 0.5\end{array}$

A) $5 e ^ { 2 }$
B) $5 e + \pi$
C) $e ^ { 2 } + \pi$
D) $5 e ^ { 2 } + \pi$

Question 46

Differentiate $\frac { 5 x ^ { 2 } } { x ^ { 3 } + 1 }$ .

A) $\frac { 25 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$
B) $\frac { 5 x \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$
C) $\frac { 5 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$
D) $\frac { 10 x } { 3 x ^ { 2 } + 1 }$

Question 47

Find the second derivative of $f ( x ) = e ^ { - x ^ { 2 } }$ at x = 1.5.Round to three decimal places.

A)0.738
B)-0.105
C)0.316
D)-1.159

Question 48

Determine the derivative rule for finding the derivative of the reciprocal function: $\frac { 1 } { g ( x ) }$

A) $\frac { - g ^ { \prime } ( x ) } { [ g ( x ) ] ^ { 2 } }$
B) $\frac { 1 } { g ^ { \prime } ( x ) }$
C) $[ g ( x ) ] ^ { - 1 }$
D) $\frac { g ^ { \prime } ( x ) } { 2 g ( x ) }$

Question 49

Given $f ( x ) = e ^ { x }$ , $g ( x ) = 5 ^ { x }$ , and h(x)= g(x)/f(x), find $h ^ { \prime \prime } ( x )$ .

A) $5 ^ { x } e ^ { - x } ( \ln 5 + 1 ) ^ { 2 }$
B) $5 ^ { x } e ^ { - x } ( \ln 5 - 1 ) ^ { 2 }$
C) $5 ^ { x } e ^ { - x } ( \ln 5 ) ^ { 2 }$
D) $- x ( x - 1 ) 5 ^ { x - 2 } e ^ { - x - 2 } ( \ln 5 ) ^ { 2 }$

Question 50

Given $f ( x ) = e ^ { x }$ , $g ( x ) = 7 ^ { x }$ , and h(x)= g(x)f(x), find $h ^ { \prime } ( x )$ .

A) $7 ^ { x } e ^ { x } ( \ln 7 - 1 )$
B) $7 ^ { x } e ^ { x } ( \ln 7 + 1 )$
C) $7 ^ { x } e ^ { x } ( \ln 7 )$
D) $x 7 ^ { x - 1 } e ^ { x - 1 } ( \ln 7 )$

Question 51

A table of values for a function F near x = 3 and tables of values for a function G near x = 3 and near x = 7 are given below.If , estimate H'(3)using the chain rule.Use right-hand estimates for and .

Question 52

The volume of a certain tree is given by , where C is the circumference of the tree at the ground level and h is the height of the tree.If C is 4 feet and growing at the rate of 0.25 feet per year, and if h is 25 feet and is growing at 5 feet per year, find the rate of growth of the volume V (in ft³/yr).Round to 2 decimal places.

Question 53

A table of values for functions F and G near x = 3 is given below.If H(x)= F(x)/G(x), estimate H'(3)by using the quotient rule and then using right-hand estimates for and .Round to 2 decimal places.

Question 54

Differentiate .

Question 55

Find the slope of the line tangent to when x = 2.Round to two decimal places.

Question 56

The table below gives values for functions f and g, and their derivatives.
Find at x = 1.Round to 2 decimal places.

Question 57

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 2 )$ if $H ( x ) =$ [f(x)]².
$\begin{array} { c c c c c } \boldsymbol { x } & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \\0 & 1 & - 1 & 2 & 5 \\1 & - 1 & 2 & 4 & 0 \\2 & 7 & 3 & 11 & 0.5\end{array}$

Question 58

A table of values for a function F near x = 3 and tables of values for a function G near x = 3 and near x = 7 are given below.Estimate using the right-hand estimate.

Question 59

What is the instantaneous rate of change of the function at x = 2? Round to 3 decimal places.

Question 60

Find the equation of the tangent line to at x = 3 and use it to approximate the value of f(3.2).Round to 5 decimal places.

Question 61

A botanist in the field needs a quick estimate of the growth of jumping cholla cactus.In his study area, he has controlled the water and nutrients the jumping cholla receive so he can isolate the effect of sunlight on growth.He lets a function s(t)represent the average hours of sunlight per day in month t (where t = 1 is January).He lets a function g(s)represent the monthly growth of jumping cholla in centimeters at different sunlight exposures.From past experience, he quickly jots down the table below.
Suppose .
a)Find and interpret meaning of the quantity g(10).
b)Find and interpret the meaning of the quantity g'(10).
c)Find the interpret the meaning of the quantity h(10).
d)Find the interpret the meaning of the quantity h'(10).

Question 62

Find the derivative $\frac { d } { d \theta } \left( \theta \sin \left( \theta ^ { 2 } \right) \right)$ .

A) $\sin \left( \theta ^ { 2 } \right) + 2 \theta ^ { 2 } \cos \left( \theta ^ { 2 } \right)$
B) $\sin \left( \theta ^ { 2 } \right) + \theta \cos \left( \theta ^ { 2 } \right)$
C) $2 \theta \sin \left( \theta ^ { 2 } \right) + \theta \left( 2 \theta \cos \left( \theta ^ { 2 } \right) \right)$
D) $2 \theta \cos \left( \theta ^ { 2 } \right)$

Question 63

Find the derivative of $h ( x ) = \left( 4 x ^ { 3 } + e ^ { x } \right) ^ { 3 }$ .

A) $3 \left( 4 x ^ { 3 } + e ^ { x } \right) ^ { 2 } \left( 12 x ^ { 2 } + e ^ { x } \right)$
B) $3 \left( 4 x ^ { 3 } + e ^ { x } \right) ^ { 2 } \left( 12 x ^ { 2 } + x e ^ { x - 1 } \right)$
C) $3 \left( 4 x ^ { 3 } + e ^ { x } \right) ^ { 2 }$
D) $3 \left( 12 x ^ { 2 } + e ^ { x } \right) ^ { 2 }$

Question 64

Find the slope of the curve at x = 2 to the nearest whole number.

Question 65

Which is true of the following graph?

A) $g ^ { \prime } ( x ) = f ( x )$
B) $f ^ { \prime } ( x ) = g ( x )$

Question 66

Differentiate .

Question 67

Find the derivative of $g ( x ) = \sqrt { e ^ { x } + e ^ { x ^ { 5 } } }$ .

A) $\sqrt { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } }$
B) $\frac { 1 } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$
C) $\frac { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$
D) $\frac { e ^ { x } + x ^ { 5 } e ^ { x ^ { 5 } - 1 } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$

Question 68

Find the derivative of $f ( x ) = 5 ^ { 4 x + 4 }$ .

A) $( 5 \ln 4 ) 5 ^ { 4 x + 4 }$
B) $( 4 \ln 5 ) 5 ^ { 4 x + 4 }$
C) $( \ln 5 ) 5 ^ { 4 }$
D) $( 4 x + 4 ) 5 ^ { 4 x + 3 }$

Question 69

Differentiate $f ( x ) = x ^ { 9 / 10 } - x ^ { 10 / 9 } + x ^ { - 10 / 9 }$

A) $\frac { 9 } { 10 } x ^ { 4 / 5 } - \frac { 10 } { 9 } x - \frac { 10 } { 9 } x ^ { - 11 / 9 }$
B) $\frac { 9 } { 10 } x ^ { - 1 / 10 } - \frac { 10 } { 9 } x ^ { 1 / 9 } - \frac { 10 } { 9 } x ^ { - 19 / 9 }$
C) $\frac { 9 } { 10 } x ^ { - 1 / 10 } - \frac { 10 } { 9 } x ^ { 1 / 9 } + \frac { 10 } { 9 } x ^ { - 19 / 9 }$
D) $\frac { 9 } { 10 } x ^ { 4 / 5 } - \frac { 10 } { 9 } x + \frac { 10 } { 9 } x ^ { - 11 / 9 }$

Question 70

If a and b are constants and , find .

Question 71

Find $y ^ { \prime }$ when $y = a e ^ { b x } \left( 1 - e ^ { b x } \right)$ .

A) $a b e ^ { b x } \left( 1 - 2 e ^ { b x } \right)$
B) $a b e ^ { b x }$
C) $a b ^ { 2 } \left( e ^ { b x } - 2 \right)$
D) $a b e ^ { b x } \left( - e ^ { b x } \right)$

Question 72

Find a function F(x)such that and F(0)= 5.

Question 73

Differentiate $f ( x ) = \sin \left( 3 \theta ^ { 6 } + 1 \right)$ .

A) $\cos \left( 3 \theta ^ { 6 } + 1 \right)$
B) $18 \theta ^ { 5 } \cos \left( 3 \theta ^ { 6 } + 1 \right)$
C) $3 ( \ln 6 ) \theta ^ { 6 } \cos \left( 3 \theta ^ { 6 } + 1 \right)$
D) $- 18 \theta ^ { 5 } \cos \left( 3 \theta ^ { 6 } + 1 \right)$

Question 74

Differentiate $f ( x ) = e ^ { z ^ { 2 } } / \left( z ^ { 2 } - 7 \right)$ .

A) $e ^ { z ^ { 2 } }$
B) $\frac { 2 z e ^ { z ^ { 2 } } } { z ^ { 2 } - 7 }$
C) $\frac { \left( z ^ { 2 } - 7 \right) e ^ { z ^ { 2 } } - 2 z e ^ { z ^ { 2 } } } { \left( z ^ { 2 } - 7 \right) ^ { 2 } }$
D) $\frac { 2 z \left( z ^ { 2 } - 8 \right) e ^ { z ^ { 2 } } } { \left( z ^ { 2 } - 7 \right) ^ { 2 } }$

Question 75

Differentiate $f ( w ) = \sqrt { w ^ { 2 } + 8 }$ .

A) $\frac { 8 w } { \sqrt { w ^ { 2 } + 8 } }$
B) $\sqrt { 2 w }$
C) $\frac { w } { \sqrt { w ^ { 2 } + 8 } }$
D) $\frac { 1 } { 2 \sqrt { w ^ { 2 } + 8 } }$

Question 76

A particle moves in such a way that , where x is the horizontal distance the particle has traveled, in units, and t is time, in seconds.What is the average rate of change between t = 0 and ? Specify units

Question 77

Differentiate $f ( x ) = x e ^ { - 2 x }$ .

A) $e ^ { - 2 x } ( 1 - 2 x )$
B) $e ^ { - 2 x } ( 1 - 4 x )$
C) $e ^ { - 2 x } ( 1 + x )$
D) $- 2 e ^ { - 2 x }$

Question 78

Find the critical number(s)of the curve .

Question 79

Find .

Question 80

A particle moves in such a way that .What is the instantaneous rate of change at t = 0?