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1
Math 111 – Calculus I - SECTION A
SAMPLE FINAL EXAMINATION
Wednesday, December 11th, 2013 – 200 POSSIBLE POINTS
DISCLAIMER: This sample exam is a study tool designed to assist you in preparing for
the final examination in Harrison’s section A of Math 111. Although these problems are
representative of the type of questions you may be asked on the final examination, they
do not, in any way, represent an exhaustive list of the possible questions that may appear
on that test.
I.
(20 points) OBJECTIVE QUESTIONS
(1) If f and g are differentiable on the real line, f(1) = 2; f(2) = 1; g(1) = 2;
g(2) = 1; f' (1) = 2; f' (2) = 5; g' (1) = 4; and g' (2) = 6, then (g(f(1)))' = (a) 1 (b) 4
(c) 12 (d) 18 (e) 30 (f) none of the above
(2) Which of the following functions are anti-derivatives of
f(x) = sin 3 (x)cos(x) on the interval [1, 100] ?
(a)
sin 4 x
+π
4
1
(b) - ∫ sin 3 (t)cos(t)dt + e
(c) each of (a) and (b) (d) neither (a) nor (b)
x
(3) (TRUE OR FALSE) If f is differentiable at a real number a, then f is
continuous at a real number a.
(4)(TRUE OR FALSE) L’Hopital’s Rule can be applied to show that if f(x) =
cos(2x)/x, then
f ' (x) =
- 2sin(2x)
= - 2sin(2x).
1
2
II. (24 points) Compute the following limits.
x 2 − 4x
1. lim
x →4
x 2 + x - 20
3. lim
xe
x →∞
2
-3x
Tan -1 (x) + cos(x)
2. lim
x →∞
x
4. lim
x →0
+
(8x + 7)
1
x
3
III. (6 points) Below are two graphs. One is the derivative of the other. Identify which
is which.
IV. (16 points) Consider the graph of the following function f below.
GRAPH OF f
a
b
(a) Does lim f(x) exist? Is f continuous at x = a? Does lim f(x) exist? Is f continuous at x = b?
x →a
x →b
(b) Does lim- f(x) = f(a)? Does lim+ f(x) = f(a)? Does lim- f(x) = f(b)? Does lim+ f(x) = f(b)?
x →a
x →a
x →b
x →b
4
V. (30 points) In parts (1) – (5), find the derivative of the specified function/relation
with respect to x (dy/dx) (SHOW YOUR WORK – JUSTIFY YOUR ANSWERS).
(a) y = (1 + x )(1 + x ) (b) y = 2
-2
2
(d) x 5 + 3xy = 4
sin(e3x )
(x - 1) 3 (e 5x )
(c) F(x) = ln(
)
(2x + 1) 5 ( x 4 )
2
(e) y = (2x) x +3 x
5
VI. (13 points)
(1) (8 points) Use the DEFINITION of DERIVATIVE to compute the derivative of
the function f(x) = 2/(x – 3) for any x in its domain.
(2) (5 points) Utilize your work in part (1) to find the equation of the tangent line to f
at the point P = (1,-1).
6
VII. (12 points) Consider the function f(x) = (3/2)x on the interval [0,2].
(a)Approximate the integral of f over the interval using one right-hand
rectangle (in this case the rectangle is inscribed).
(b) Approximate the integral of f over the interval using 100 righthand/inscribed rectangles (HINT: Setup an appropriate Riemann sum and
enter it into your TI-89 graphing calculator. The technique you utilize for
part (b) should allow you to complete part (a)).
(c) Apply the Fundamental Theorem Of Calculus’s Evaluation Theorem
!
directly to calculate ! (3/2)x dx
7
VIII.
(24 points) Calculate the following definite/indefinite integrals exactly (YOU
MUST SHOW ALL WORK INCLUDING ALL SUBSTITUTIONS. DO
NOT APPROXIMATE USING YOUR CALCULATOR).
(1) ∫ ((5.1) + x + sec (x))dx
x
5.1
2
1
x 3 +1
(2) ∫ x e dx
0
2
6
(3) ∫
1
t dt
(4) ∫ sin 3 ( x) dx
t +3
8
IX.
(20 points) Consider the following function.
f(x) = e!(x-­‐1)
!
(1) Find the domain of f. Show that f is positive on its domain.
(2) Find the horizontal asymptote(s) of f.
(3) 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.
(4) Determine any inflection points of f. Find intervals where (i) f is concave
upward, (ii) f is concave downward.
(5) Use your previous answers to sketch f.
9
(WORK PAGE)
10
X.
(35 points) Application Problems -- complete exactly FOUR of the FIVE
problems -- SHOW YOUR WORK to OBTAIN FULL CREDIT – Fifth
Problem is a bonus problem for up to NINE POINTS.
(1) An inverted conical water tank with a height of 12 feet and a radius of 6 feet
drains through a hole in the vertex at a rate of 2 ft3/s. What is the rate of change
of the height of the water in the tank when the water is 3 feet deep? NOTE: The
volume of a cone is given by V = 1/3(πr2h) where V is the volume of the cone, r is
the radius of the cone, and h is the height of the cone.
(2) The aerobic rating of a person x years old is modeled by the function
A(x) = largest?
!!"(ln(x)-­‐2) x
for x > 10. At what age is a person’s aerobic rating the
(3) Assume a stone is thrown vertically upward with an initial velocity of 64 ft/s
from a bridge 96 feet above a river. What is the maximum point above the
river reached by the stone? With what velocity will the stone strike the river?
(4) An oral painkiller is administered to a patient. After t hours, the concentration
of drug in the patient’s bloodstream is given by C(t) = 2t/(3t2 + 16) mg. At
what rate is the concentration of drug changing after the first hour (in mg/hr)?
Is the drug concentration changing at an increasing rate or a decreasing rate at
this time (EXPLAIN)?
(5) Assume the length L(t) of a certain organism at time t (t is in years) satisfies
the following differential equation
dL/dt = e-0.1t where t > 0
Assume that limt→∞ L(t) = 25
(a) Find the function L(t).
(b) Compute L(0) (the initial length of the organism).
11
(WORK PAGE)
12
(WORK PAGE)