MTH 131 Final Exam Test Prep Questions
Fall 2013 – Johnson
1. The graph of a function is formed by vertically stretching the graph of
by a factor of
, and shifting it to the left units and down unit. Find an equation for function and graph
it for
and
.
2. Find
, and find
(a)
(b)
(c)
3. Find
, and find
(a)
(b)
(c)
(d)
(e)
4. Find the indicated derivatives
(a)
(b)
(c)
(d) Find
if
(e) Find
for
(f)
(g)
(h)
(i) Find
for
(j) Find
for
(k) Find
for
(l) Find
for
(m)
5. Find the equation(s) of the tangent line(s) to the graphs of the indicated equations at the
point(s) with the given value of :
(a)
(b)
;
;
6. A company is planning to manufacture and market a new two-slice electric toaster. After
conducting extensive market surveys, the research department provides the following
estimates: a weekly demand of
toasters at a price of
per toaster and a weekly demand
of
toasters at a price of
per toaster. The financial department estimates that weekly
fixed costs will be
and variable costs (cost per unit) will be .
(a) Assume that the relationship between price and demand is linear. Use the
research department’s estimates to express as a function of and find the domain
of this function.
(b) Find the revenue function in terms of
and state its domain.
(c) Assume that the cost function is linear. Use the financial department’s estimates to
express the cost function in terms of .
(d) Graph the cost function and revenue function on the same coordinate system for
. Find the break-even points and indicate regions of loss and profit.
(e) Find the profit function in terms of .
(f) Evaluate the marginal profit at
and
and interpret the results.
7. Summarize the pertinent information (domain, intercepts, asymptotes, critical values, local
extrema, and inflection points) and then graph:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
8. A company manufactures and sells
cost equations are, respectively,
digital cameras per week. The weekly price-demand and
(a)What price should the company charge for the cameras, and how many cameras
should be produced to maximize the weekly revenue? What is the maximum revenue?
(b) What is the maximum weekly profit? How much should the company charge for the
cameras, and how many cameras should be produced to realize the maximum weekly
profit?
9. A commercial cherry grower estimates from past records that if
trees are planted per acre,
then each tree will yield an average of
pounds of cherries per season. If, for each additional
tree planted per acre (up to ), the average yield per tree is reduced by 1 pound, how many
trees should be planted per acre to obtain the maximum yield per acre? What is the maximum
yield?
10. For
, find
and
11. For
, find:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
12. Find
,
, and
for:
(a)
(b)
13. Find all critical points and test for extrema for
14. A company produces units of product and units of product (both in hundreds per
month). The monthly profit equation (in thousands of dollars) is given by
(a) Find
and interpret the results.
(b) How many of each product should be produced each month to maximize profit? What is
the maximum profit?
15. A rectangular box with no top and six compartments (see figure) is to have a volume of
inches. Find the dimensions that will require the least amount of material.
16. Partition
using
17. Integrate:
(a)
(b)
(c)
(d)
(e)
(f)
into five equal subintervals on
cubic
and calculate the Riemann sum
(g)
(h)
(i)
(j)
(k)
(l)
18. Find a function that
that satisfies both conditions:
19. Find the average value of
in the same coordinate system.
over the interval
then graph
and its average
20. Find the area bounded by the graphs of the indicated equations over the given interval:
(a)
(b)
(c)
;
;
,
,
;
21. The shelf life (in years) of a laser pointer battery is a continuous random variable with
probability density function
(a) Find the probability that a randomly selected laser pointer battery has a shelf life of
3 years or less
(b) Find so that the probability of a randomly selected laser pointer battery lasting
years or less is
(c) Graph
for
and show the shaded region for part (a)
22. For a continuous income stream with rate of flow
(a) Find the total income produced in the first two years
(b) Find the future value, at 2.95% interest, compounded continuously for the first two
years
23. For the price-demand equation
, and the price-supply equation
, find (to the nearest integer) and graph:
(a) The equilibrium price level
(b) The consumers’ surplus at the equilibrium price level
(c) The producers’ surplus at the equilibrium price level