Math 142 Final Review with key
1. A company finds the daily demand for its cookies is a linear function of price. If price increases by $0.50 per lb, demand decreases by 5 lbs. The daily demand when price is $5 per lb is 30 lbs. It costs them $60 per day for labor and an additional $1.00 per lb for ingredients.
a) Find the profit function.
b) Find the marginal profit.
c) Find the quantity and price for maximum profit.
2. Find the domain and all asymptotes for each function.
a) 
b) 
c) f(x)=ln(2x + 6)
3. Solve for x:
a) 
b) ln(5x-2) – 2ln(x)=ln2
c) ln(x+2) + ln(x-2)=1
4. What transformations of f(x) result in g(x)?
a) 
b) 
5. Find the amount after 10 years if $10,000 is invested at 8% compounded
a) quarterly b) continuously c) How long will it take the amount to double if compounded continuously?
6. Find the present value of $30,000 ten years from now if money earns 7% compounded continuously.
7. The data represents the concentration in ppm of lead in the air in a certain location.
1985 1986 1987 1988 1989 1990 1991 1992
.266 .153 .105 .087 .071 .058 .049 .043
Compare exponential and quartic regression models. Which is a better fit of given data? Which would be better for predicting 1996 data?
8. The data represents the total number of flu caused deaths in the weeks after the flu epidemic of 1918.
Wk2 wk4 wk6 wk8 wk10 wk12
517 6528 37,853 73,477 86,957 93,641
View the data in a stat plot. What model might be the best fit. Find the equation for this model, the inflection point and limiting value.
9. Find all discontinuities.
a) 
b) 
10. Find the points where f is not differentiable.
a) 
b) 
c 
11. Find the derivatives and determine where the tangent line is horizontal.
a) 
b) 
c) 
d) 
e) 
Use log rules. Do not
solve for horizontal tangent line, or use the calculator.
f) 
12. Find the equation of the tangent line at x=a.
a) 
a=3 b) 
a=-2
13. Demand for a commodity is x=600-2p. The cost equation for the same commodity is C(x)=120x+10,000.
a) Find the revenue as a function of x and find its domain.
b) Find the profit function.
c) What is the maximum profit and at what quantity does it occur?
d) Approximate the additional profit if the quantity produced and sold increases from 160 to 161. Use the marginal profit.
e) Find the equation of the tangent line at x=160.
f) Approximate the profit at x=161 using the tangent line.
g) Find the marginal average cost at x=160.
14. Find all local extrema, inflection points and asymptotes of each.
a) 
b) 
c) d) 
e) 
f) f(x)= xln(x)
15. 11-14 and 17-22 on pages 352 and 353
16. Find 
and write it in terms
of x.
a) y=
w= ln(1+u) u= 
b) 
w= 
u= -0.02x
17. Find the antiderivatives:
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
i) 
18. A marginal
average cost function is 
. The total cost producing
1000 units is $65,000.
a) Find the cost function.
b) Find the average value of the cost function for x between 0 and 1000.
c) Find the average cost per unit at x=1000 units.
19. Find the antiderivative that satisfies the given condition.
a) 
f(e)=2 b) 
f(1)=2.5
20. Find the
Riemann sums for 
, using 6 equal intervals on [0,3]
a) and the midpoint of each interval.
b) and the left endpoint of each interval.
c) and the right endpoint of each interval.
How do these compare to 
?
21. Show that 
is an antiderivative for 
. Use the fundamental theorem of calculus to find the exact
value of 
.
22. Find the area between the curves over the given interval.
a) 
[-3, 4]
b) 
[-2, 5]
23. Money is deposited into an account as a continuous income stream at the rate of
.
a) Find the total amount deposited after 5 years.
b) How much is in the account after 5 years if all money earns interest at 5% per year compounded continuously? How much of this is interest?
24. Find the consumer’s surplus and the producer’s surplus at equilibrium if the demand price is D(x)= 100 – 0.2x and the supply price is S(x)= 25 + 0.05x.
25. pg 488 1-22
Key:
1. a) P(x)=
b) 
c) 35 lbs at $4.50
per lb.
2.a) domain=
vertical asymptotes
x=1and x=3 horizontal as. y=0.
b) domain is all real numbers. Horizontal asymptote y=0 for x tending to minus infinity
c) domain=
vertical as. x= -3
3. a) 
b) ˝ or 2 c) 
4. a) Shift right 1 unit, vertically expand by a factor of 3, shift up 5
b) Shift left 1 unit, vertically expand by a factor of 2, shift down 7 units
5. a) $22080.40 b) $22255.41 b) 8.664 years or about 8 years, 8 months
6. $14,897.56
7. The quartic is a better fit of the given data. The exponential model continues the downward trend where the quartic increases after 1992. Without new data, the exponential would be better for predicting the 1996 data.
8. Logistic. Inflection pt (6.466, 46110.629) limiting value 92221.2576
9. a) x=1 and x= -1 b) x=3
10. a) x=0 b) x=1, x=3 c) differentiable for all x.
11. a) 
horizontal tangent at
x= -2
b)
horizontal tangent
at x=e
11 c) 
horizontal tangent
at x= 0, 1, -1
d)
no horizontal
tangents
e) 
f) 
horizontal tangent at x=0
12. a) 
b) 
13. a) R(x)=
b)
c) $6200 at x=180
d) 20
e) y = 20(x-160) + 6000
f) 6020
g) -0.39
14. a) no local extrema, VA x=2, HA y=2/3
b) VA x=2 and x= -2, HA y=2 local max at x=0, no inflection points
c) HA y=0, no VA, local min at x= -2, inflection at x= -4
d) VA x=0, HA y=0, local min at x=1, no inflection points
e) VA
x=0, HA y=0, local max at x=e, inflection at x=
f) No VA or HA, local min at 1/e, no inflection points
16. a) 
b) 
17. a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
i) 
18.a) C(x) = 25000 + 40x b) $45000 c) $65
19.a) ln|x| + 1 b) 
20. a) 11.9375 b) 9.875 c) 14.375 d) The actual integral is 12.
21. 
22. a) 33.5 b) 160.25
23. a) $10,517.092 b) $11,923.63 $1406.54 is interest
24. CS=$9000 PS=$2250