ch6

1.Find the linear approximationL(x)

toy=f(x)

nearx=a

for the function.

f(x) =1

x

,a=5

L(x) =

2.Find the linear approximationL(x)

toy=f(x)

nearx=a

for the function.

f(x) = sin(x),a=

2

L(x) =

3.Find the differential of the function.

y=

2+x

dy=

4.For the function below, write the differential.

f(x) = ln(1+x²)

dy=dx

Calculatedyfor the given values ofxanddx.

x=2anddx= 0.2

dy=

How accurate isdyin approximatingy(i.e., what is their difference) to the nearest thousandth?

5.Find the change in volumedV(in units³) if the sides of a cube change from10to10.1.

dV=

units³

6.A spherical ball is measured to have a radius of10mm,with a possible measurement error of0.1 mm.What is the possible change in volume (in mm³)?

mm³

7.The stopping distance for an automobile isF(s) = 1.1s+ 0.054s²

ft, wheresis the speed in mph. Use the Linear Approximation to estimate the change in stopping distance per additional mph whens=25.

ft

Use the Linear Approximation to estimate the change in stopping distance per additional mph whens=35.

ft

7.Write the domain of the functionf.

f(x) =2x³+3

x

x0

x3

(0,)

(, 0)

(,)

Find thex-values of all critical points. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x=

8.Write the domain of the functionf.

f(x) =16x

49x²+ 1

x1

7

x7

(0,)

(, 0)

(,)

Find thex-values of all critical points. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x=

9.Find the critical points in the domain of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

y= ln(x9)

x=

10.Find the critical points in the domain of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

y=x^3/27x^5/2

x=

11.Consider the functionf(x) = 2x³9x²24x+ 9

on the interval[2,5].

What is the absolute minimum off(x) on[2,5]?

What is the absolute maximum off(x) on[2,5]?

12.Find the critical points off(x) =4sinx+4cosx

and determine the extreme values on0,

2

.

(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

critical pointsx=

maximumf(x) =

minimumf(x) =

13.Find the local and absolute minima and maxima for the function over(,).

(Order your answers from smallest to largestx.)

y= 3x⁴+12x³24x²

(x,y)=

both local and absolute minimum

(x,y)=

---Select---

local minimum

local maximum

absolute minimum

absolute maximum

both local and absolute minimum

both local and absolute maximum

(x,y)=

---Select---

local minimum

local maximum

absolute minimum

absolute maximum

both local and absolute minimum

both local and absolute maximum

14.A ball is thrown into the air, and its position is given byh(t) =4.9t²+62t+6m,

wheretis the time in seconds since the ball was thrown. Find the height (in meters) at which the ball stops ascending.

m

How long (in seconds) after it is thrown does this happen?

s

15.Consider the production of gold during the California gold rush(1848-1888).The production of gold can be modeled byG(t),

wheretis the number of years since the rush began(0t40),

andGis ounces of gold produced (in millions). A summary of the data is shown below.

G(t) =(25t)

(t²+ 16)

Find the year that the maximum (global) gold production occurred.

What was the amount of gold (in millions of ounces) produced during this maximum? (Round your answer to one decimal place.)

million oz

Find the year that the minimum (global) gold production occurred.

What was the amount of gold (in millions of ounces) produced during this minimum?

million oz

2.Determine over what interval(s) (if any) the Mean Value Theorem applies. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

y=1

x³

^3.Determine over what interval(s) (if any) the Mean Value Theorem applies. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

y=

x²36

4.Determine over what interval(s) (if any) the Mean Value Theorem applies. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

y= ln(5x7)

5.Graph the function on a calculator and draw the secant line that connects the endpoints. Estimate the number of pointscsuch thatf'(c)(ba) =f(b)f(a).

y=5x³+2x+1over [1, 1]

points

6.Use the Mean Value Theorem and find all points0 <c< 2

such thatf(2)f(0) =f'(c)(20).

f(x) = (x1)¹⁶

c=

7.(a) Show there is nocsuch thatf(1)f(1) =f'(c)(2).

f(x) =1

x²

Assume there exists ac, such thatf(1)f(1) =f'(c)(2),

or equivalentlyf'(c) =f(1)f(1)

2

.

Given the functionf(x) =1

x²

,

we havef(1) =,

f(1) =,

andf'(x) =

sof'(c) =.

Thenf(1)f(1)

2

=,

but setting this value equal to the expression forf'(c)

leads to an inconsistent equation. Thus, there is no solutioncsuch thatf(1)f(1) =f'(c)(2).

(b) Explain why the Mean Value Theorem does not apply over the interval[1, 1].

(Select all that apply.)

The Mean Value Theorem does apply.

The Mean Value Theorem does not apply since the function is discontinuous at a point.

The Mean Value Theorem does not apply since the function is not differentiable at a point.

The Mean Value Theorem does not apply since thef(b) does not exist.

The Mean Value Theorem does not apply since thef(a) does not exist.

8.Determine whether the Mean Value Theorem applies for the function over the given interval[a,b].

Justify your answer. (Select all that apply.)

y=3+ |x| over [1, 1]

The Mean Value Theorem does apply.

The Mean Value Theorem does not apply since thef(b) does not exist.

The Mean Value Theorem does not apply since thef(a) does not exist.

The Mean Value Theorem does not apply since the function is not differentiable at a point.

The Mean Value Theorem does not apply since the function is discontinuous at a point.

9.Determine whether the Mean Value Theorem applies for the function over the given interval[a,b].

Justify your answer. (Select all that apply.)

y= ln(x+6) over [0,e6]

The Mean Value Theorem does apply.

The Mean Value Theorem does not apply since thef(a) does not exist.

The Mean Value Theorem does not apply since thef(b) does not exist.

The Mean Value Theorem does not apply since the function is discontinuous at a point.

The Mean Value Theorem does not apply since the function is not differentiable at a point.

10.Determine whether Rolle's Theorem can be applied to the function on the given interval; if so, find the value(s) ofcguaranteed by the theorem. (Enter your answers as a comma-separated list. If Rolle's Theorem does not apply, enter DNE.)

f(x) =

x

(4x) on [0,4]

c=

11.Examine the graph shown. Estimate the valuecguaranteed by the Mean Value Theorem, and alsof'(c),

on the intervals:

[5, 1]; [1,7]; [5, 9].

On[5, 1]

f'(c)

=

c=

On[1,7]

f'(c)

=

c=

On[5, 9]

f'(c)

=

c=

2.Findy'

andy''

forx²+8xy3y²=2.

y'

=y''

=

4.Supposeyis implicitly defined as a function ofxby the equations below. Find the slope of the tangent line at the point(1, 1).

(a)8x

y

y²=7

x

dy

dx

=

(b)8+ye^y=8x+e^x

dy

dx

=

1.Use the information in the table to findh'(a)

at the given value ofa.

h(x) = (1 +g(x))³;a= 2

xf(x)f'(x)g(x)g'(x)04502152

3028841

377

27

h'(a) =

2.The total cost to producexboxes of cookies isCdollars, whereC= 0.0001x³0.04x²+4x+400.

Intweeks,production is estimated to bex=1300+ 100tboxes.

(a)

Find the marginal costC'(x).

C'(x) =

(b)

Use Leibniz's notation for the chain rule,dC

dt

=dC

dx

dx

dt

,

to find the rate with respect totimetthat the cost is changing.

dC

dt

=

(c)

Use the results from part (b) to determine how fast costs are increasing (in dollars per week) whent=4weeks.

dollars per week

4.Find the requested higher-order derivative for the given function.

d²y

dx²

ofy=5sin(x) +x²cos(x)

d²y

dx²

=