Math 6 Name Exam #3: Chapters 5 and 6 All problems are worth 15 points each. You must show all work to receive full credit.

1. Find the intervals where /(x) is increasing, the intervals where /(x) is decreasing, and the local extrema.

f(x) =x' +2x' +5 (+1r civ) e ^ V) < ^ V >

- 4 o

Intervals /(A-) is decreasing Intervals /{x) is increasing

Local extrema: X ? : > H XQCC^ ("'^J^)

2. A company estimates that it will sell N(x) units of a product after spending $x thousand on advertising, as given by

N{x) - -2x' + 90x' - 750x + 2000 for 5 < x < 25.

a. When is the rate of change of sales, N '(x), increasing? Decreasing?

< \Inc. I'S 0(0.

Huh '^A+m^r

aid dtmat\r C\5/X5^

b. Find the inflection points for the graph of A .

K(l5)- -2(i^J?'-'^C(i5T-'^3^i5) vzcOD= 4250 (J 5,42^50)

c. Find the intervals where the graph of N is concave up or concave down.

3. Circle the limit(s) that cannot use L'Hopital's rule. Note; there may be more than one.

lim ~

^^«ln(l + e~") Q

4. Use L'Hopital's rule to evaluate the following limits.

„ m ^ ^ = « \ j ! S _ _ VvO\ J r - ^

5. In spite of the prediction of a paperless computerized office, paper and paperboard production in the United States has steadily increased. In 1990 the production was 80.3 million short tons, and since 1970 production has been growing at a rate given by

/ '(A) = 0.048^ + 0.95, where x is years after 1970. Find /(x), and the production levels in 2000.

)*3 = M^+C

Jfxzcoo tn2CQO

6. The marginal price of a supply level of J: bottles of baby shampoo per week is given by , 300

^^^^ = p ^ W Find the price-supply equation i f the distributor of the shampoo is willing to supply 75 bottles a week at a price of $ 1.60 per bottle. ^

3A+25

7. Calculate the definite integrals by referring to the figure with the indicated areas.

Aiea A = 3.5 eh

a. •5.J

b. ^fix)dx =z J 0

2-

c.

d, f{x)dx ^ -.JO-

Aiea B = 12

8. Assume we know the following. Then calculate the definite integrals.

^ ^

X ax - —. 0 2

X dx -9, and r4 37

X dx = —.

r3 a. 4x'

1-3 b. Qx'-2x)dx

q ^

r3 C.

-51 5

d. — — — J

(•4 e. 3.x' dx

9, From past records a management service detennined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by

M'(^') = 90x'+5000, where M{x) is the total accumulated cost of mamtenance for x years. Write a definite integral that will give the total maintenance cost from the beginning of the second year to the end of the seventh year. Evaluate the integral.

10. Using production and geological data, the management of an oil company estimates that oil will be pumped from a producing field at a rate given by

R'(l) = } ^ + 3 0<t<20 t +1

where R'(t) is the rate (in thousands of barrels per year) / years after pumping begins. Approximately how many barrels of oil will the field produce during the first 5 years of production?

m = llmL f 3 at. -fJ^dt fff3 cii

da- 2-t- =y dt=4kk, Tf t = 5, 5 1= < '

(,1 -y/ 2&

•2^

2/0/5 terf^'^ 4 ^'