ch7

7.Consider the function. (If an answer does not exist, enter DNE.)

f(x) =x³4x²+x+2

(a)

Determine intervals wherefis concave up or concave down. (Enter your answers using interval notation.)

concave up

concave down

(b)

Determine the locations of inflection points off. (Enter your answers as a comma-separated list.)

x=

8.Determine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. Enter EMPTY orfor the empty set.)

f(x) = (x6)(4x³)

concave upconcave down

Find the points of inflection. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)points of inflectionx=

9.Determine the intervals on which the function is concave up/down.

f(x) =8xe^x2

Concave up:

,

3

2

0,

3

2

3

2

, 0

3

2

,

3

2

,

,

3

2

never

Concave down:

,

3

2

0,

3

2

3

2

, 0

3

2

,

3

2

,

,

3

2

never

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

x=

10.Consider the function. (If an answer does not exist, enter DNE.)

f(x) =x+x²x³

(a)

Determine intervals wherefis increasing or decreasing. (Enter your answers using interval notation.)

increasingdecreasing

(b)

Determine the local minima and maxima off. (Enter your answers as comma-separated lists.)

locations of local minimax=locations of local maximax=

(c)

Determine intervals wherefis concave up or concave down. (Enter your answers using interval notation.)

concave upconcave down

(d)

Determine the locations of inflection points off. (Enter your answers as a comma-separated list.)

x=

14.Find the first and second derivatives of the function.

f(x) =x

7+x²

f'(x)=f''(x)=

Identify the intervals on which it is concave up/down.

Concave up:

(

21

, 0)(

21

,)

(

7

, 0)(

7

,

21

)

(,

7

)(

7

,

21

)

(,

21

)(0,

21

)

none of the above

Concave down:

(

21

, 0)(

21

,)

(

7

, 0)(

7

,

21

)

(,

7

)(

7

,

21

)

(,

21

)(0,

21

)

none of the above

Determine all local extrema using the second derivative test. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)local maximax=local minimax=

4.Evaluate the limit atinfinity. (If an answer does not exist, enter DNE.)

lim

t

t

16t²+3

8.Evaluate the limit.

lim

x

x²2x+7

x+3

9.Evaluate the limit.

lim

x

x⁴8x³+ 1

77x²9x⁴

11.Find the horizontal and vertical asymptotes. (Letnrepresent an arbitrary integer. Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) =1

36x²

horizontaly=verticalx=

12.ind the horizontal and vertical asymptotes. (Letnrepresent an arbitrary integer. Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) =x²+6

x²+1

horizontaly=1

1

verticalx=

13.Find the horizontal and vertical asymptotes. (Letnrepresent an arbitrary integer. Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) = cos(x) + cos(7x) + cos(9x)

horizontaly=verticalx=

6.Consider a limousine that getsm(v) =(1202v)

3

mpg at speedv(in mph), the chauffeur costs$17/h,and gas is$3.5/gal.

Find the cost (in dollars) per mile at speedv.

C(v) =

Find the cheapest driving speed (in mph). (Round your answer to two decimal places.)

mph

7.Consider two nonnegative numbersxandysuch thatx+y=18.

Find the values ofxandythat optimize the following quantity. (If an answer does not exist, enter DNE.)

xy

Maximize the quantity.

{x,y} =

Minimize the quantity. (Order your answers from smallest to largestx, then from smallest to largesty.)

{x,y}=

{x,y}=

9.A window is composed of a semicircle placed on top of a rectangle. If you have20ftof window framing materials for the outer frame, what are the dimensions that maximizes the area (in ft²) of the window you can create? (Round your answers to two decimal places.)

radius of semicircleftwidth of rectangleftheight of rectangleft

10.You are the manager of an apartment complex with 50 units. When you set rent at$900/month,all apartments are rented. As you increase rent by$25/month,one fewer apartment is rented. Maintenance costs run$50/monthfor each occupied unit. What is the rent (in dollars) that maximizes the total amount of profit?

$

2.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=35.

ft

Use the Linear Approximation to estimate the change in stopping distance per additional mph whens=45.

ft