Exam # 2
Fall 2005
MATH 1100-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) Use (non-graphing) calculators only on the problems marked with c .
(5) A list of definitions and formulas is attached at the end.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1. (a) Use implicit differentiation to find
dy
, if 2exy − y = 3.
dx
(b) Find the slope of the tangent line at the point (ln 2, 1). Simplify your answer!
3
Problem 2. Consider the function f (x) = ln
s
3
x2 + 1
.
1−x
(a) Use the properties of logarithms to write the function f (x) as a sum, difference, or multiple
of logarithms.
(b) Find f ′ (x).
4
c Problem 3. (a) How much should be deposited in an account paying 5% interest compounded
monthly in order to have a balance of $40,000 five years from now?
(b) How much should be deposited if it is compounded continuously?
5
Problem 4. The demand function for a product is given by
p = 60 − 0.04x, 0 ≤ x ≤ 1500.
(a) Find the price elasticity of demand when x = 100. Interpret your result.
(b) Find the values of x and p that maximize the total revenue. Check your answer with the
second derivative test.
6
Problem 5. 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 48 inches. Find the dimensions of the box that will have
the maximum volume, assuming the base of the box is a square.
7
Problem 6. Find the following limits:
(a) lim+
x→2
(b) lim
2x
=
−4
x2
2x
=
−4
x→∞ x2
2x2
=
x→∞ x2 − 4
(c) lim
8
Problem 7. Consider the function: f (x) =
(a) x-intercept(s) and y-intercept(s).
(b) Vertical asymptote(s).
(c) Horizontal asymptote(s).
2x
. Find the following:
x2 − 4
9
Problem 8. Consider the function: f (x) =
f ′ (x) = −
2(x2 + 4)
<0
(x2 − 4)2
2x
. It has
x2 − 4
and
(a) Fill in the sign table.
x
f ′ (x)
f ′′ (x)
f (x)
(b) The interval(s) on which f is concave upward.
(c) The interval(s) on which f is concave downward.
(d) Inflection point(s).
f ′′ (x) =
4x(x2 + 12)
.
(x2 − 4)3
10
Problem 9. (a) Using the problems 7 and 8 sketch the graph of the function f (x) =
c (b) Sketch the graph of the function g(x) = ln(x + 1).
2x
.
x2 − 4
11
Problem 10. Consider the function f (x) = xe−2x .
(a) Find the derivative of f .
(b) Find the extrema points on the interval [0, 1].
12
Useful definitions and formulas
(1) Price elasticity of demand is given by η =
p/x
, where p is the price per unit and x is the
dp/dx
number of units produced (or sold).
(2) Compound interest formulas:
r nt
,
if it is compounded n times per year;
A= P 1+
n
A = P ert ,
if it is compounded continuously.
Here P =the amount deposited, t=number of years, A=the balance and r=the annual interest rate (in decimal form).
(3)
d x
d u
du
[e ] = ex and
[e ] = eu · .
dx
dx
dx
(4)
d
1
d
1 du
[ln x] = and
[ln u] = · .
dx
x
dx
u dx
(5) ln ex = x and eln x = x.
(6) ln xy = ln x + ln y, ln xn = n ln x and ln
x
= ln x − ln y.
y