Download Sample Exam III

Math 111 – Calculus I
Sample Exam #3 – Fall 2007
Tuesday/Thursday, November 6th/8th, 2007
I.
TRUE OR FALSE/MULTIPLE CHOICE (21 points)
1. A continuous function f on a closed interval [a,b] attains its global maximum
value at exactly one real number x on its domain (i.e. the interval [a,b]).
2. Assume f is a differentiable function on [0,1] such that f(0) = 0 and f(1) = 3.
Then, there exists a c in [0,1] such that f’(c) (a) is equal to 0 (b) DNE (c) is
equal to 3 (d) is equal to 6 (e) none of the above.
3. The only critical value of the function f(x) = x1/2(2x – 6)2 is at x = 3.
II.
Limits/Derivatives of Hyperbolic/Inverse Hyperbolic Functions (24
points)
1.
Show that for any x in the domain of the hyperbolic tangent function, 1 –
tanh2(x) = sech2(x).
2.
Compute lim 2 tanh( 3x)
3.
Compute the derivative of the following relation with respect to x.
x 
sinh-1 (y) + tanh (x2y) = 1
III.
Analyzing and Graphing Functions (31 points)
Consider the following function. JUSTIFY YOUR ANSWERS IN EACH PART OF THE PROBLEM.
f(x) 
1
1 x2
(a) (6 points) Find the domain of f. Is f an odd function/even function? Show that f is positive on its
domain.
(b) (6 points) Find the horizontal asymptote(s) of f.
(c) (6 points) Determine any critical points of f. Determine any local maximum or minimum values
of f. Find intervals where (i) f is increasing, (ii) f is decreasing.
(d) (6 points) Determine any inflection points of f. Find intervals where (i) f is concave upward, (ii) f
is concave downward.
(e) (7 points) Use your previous answers to sketch f.
IV.
Application Problems (24 points)
1.
Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where 0 <
x < 4 and y = (40 – 10x)/(5 – x). Your profit on a grade A tire is twice your profit on a grade B tire.
What is the most profitable number of each kind to make?
2.
A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of
its base is twice the width. Material for the base costs $5 per square meter. Material for the sides costs
$10 per square meter. Find the cost of materials for the cheapest such container (NOTE: The volume
of a rectangular container is given by V = lwh where l is the length, w is the width, and h is the height
of the rectangular container respectively).
3.
At 2:00 PM, a car’s speedometer reads 20 mi/hr. At 2:10 PM, it reads 60 mi/hr. Show that at some
time between 2:00 PM and 2:10 PM the acceleration is exactly 240 mi/hr2.
V.
Calculating the Limit of a Function (L’Hopital’s Rule – Sullivan Section
only)
Find the limit of the following functions.
1. lim
x 1
1
1
 2
ln(x) x  1
2. lim (sin(x))
x 0
x