Review problems for Exam #2
Math 1100, Fall 2005
(1) Find the derivative of the following functions:
2
(a) (3x2 + x)e−x .
3
(b) (e2x
r+ 1) .
x−1
(c) ln
.
x2 + 1
(d) (ln x3 )2 .
2
ex
(e) ln
.
1 + ex
(2) Find
dy
implicitly:
dx
(a) yex + 2xy 2 + xe−x = 1.
(b) ln(3xy 2 ) − xy + x2 = 3x.
(3) For the following functions find:
(i) The x an y intercepts.
(ii) The vertical asymptote(s); compute the limits: lim+ f (x) and lim− f (x), where x = c
x→c
x→c
is the vertical asymptote.
(iii) The horizontal asymptote(s); compute the limits: lim f (x) and lim f (x).
x→∞
(iv)
(v)
(vi)
(vii)
(viii)
x→−∞
The relative extrema.
The absolute extrema on the interval [−2, 5].
The intervals where the graph is concave downward and concave upward.
The inflection points.
Then sketch the graph.
(a) f (x) = x4 − 4x2 + 4.
2x2 − 10
(b) f (x) =
.
x+3
3x
(c) f (x) =
.
x+2
4x
(d) f (x) =
.
9 − x2
5
(e) f (x) = x − 5x4 .
(f) f (x) = x2 e−x .
(g) f (x) = x ln x.
(4) Sketch the graph of the function f that has the following properties:
(a) f ′ (1) = 0 and f ′ (3) does not exist.
(b) f ′ (x) > 0 for 1 < x < 3.
(c) f ′ (x) < 0 for x < 1 and x > 3.
(d) f ′′ (x) < 0 for x < 3 and f ′′ (x) > 0 for x > 3.
(e) lim f (x) = 2.
x→∞
(f) lim f (x) = −∞.
x→−∞
1
2
(5) A page of a book is to have an area of 90 square inches, with 1 inch margins at the bottom
and sides and a 1/2 inch margin area at the top. Find the dimensions of the page which will
allow the largest printed area.
(6) A real estate company owns 180 apartments which are fully occupied when the rent is $300
per month. The company estimates that for each $10 increase in rent, 5 apartments will
become unoccupied. What rent should be charged in order to maximize the revenue?
(7) Find two positive real numbers whose sum is 40 and whose product is maximum.
(8) A package can be send by parcel post only if the sum of its length and girth (the perimeter
of the base) is not more than 96 inches. Find the dimensions of the box that will have the
maximum volume, assuming the base of the box is a square.
(9) The demand function for a product is given by p = 30 − 0.05x, 0 ≤ x ≤ 100.
(a) Find the price elasticity of demand when x = 10. Is is elastic, inelastic, or it has unit
elasticity?
(b) Find the values of x and p that maximize the total revenue.
(10) How much should be deposited in an account paying 5% interest compounded semianually in
order to have a balance of $10, 000 four years from now?
(11) Find the balance A (in dollars) in a saving account if $1000 are deposited at the rate of 3%,
compounded continuously.
(12) The cost of producing x units of a product is modeled by C = 100 + 50x − 50 ln x, x ≥ 1.
(a) Find the average cost function C.
(b) Find the minimum average cost.