Exam # 2
Fall 2005
MATH 1100-05
Instructor: Oana Veliche
Time: 1 hour
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. Find the derivative of the function f (x) = (x2 + 1)e1−x . Simplify your answer.
3
Problem 2. Consider the function f (x) = ln
r
3
2x
.
+1
x3
(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 8% interest compounded
monthly in order to have a balance of $30,000 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
c Problem 5. A rectangular page is to contain 30 square inches of print. The margins at the top
and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide.
Find the dimensions of the page such that the least amount of paper is used.
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)
(x2 − 4)2
2x
. It has
x2 − 4
and
f ′′ (x) =
(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).
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
dy
, if
dx
2exy − y = 3.
Problem 10. (a) Use implicit differentiation to find
(b) Find the slope of the tangent line at the point (ln 2, 1). Simplify your answer!
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