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APPM 1340
Final Exam
Fall 2012
On the front of your bluebook, please write: a grading key, your name, student ID, section, and
instructor’s name (Chang or Dai). This exam is worth 150 points and has 11 questions. Show all work!
Answers with no justification will receive no points. Simplify all answers unless otherwise instructed. No
notes, calculators, or electronic devices are permitted.
1. (32 points) Evaluate the following expressions.
√
4x + 1 − 3
1
1
(a) lim
(c) lim
−
x→2
x−2
|x|
x→0− x
(b) lim θ cot(θ/3)
θ→0
(d) lim
t→−4
1
4
+ 1t
4+t
1
.
4+t
2. (10 points) Use the definition of derivative to differentiate f (t) =
3. (12 points) Differentiate the following functions. Leave your answers unsimplified.
(a) f (x) = cos(sin3 x)
√
2
(b) g(x) = x3 + 3 x2 + 1
4. (8 points) Let a and b be constants, and b 6= 0. Find the numbers at which the following function is
discontinuous.


x<0
x + 2,
2
f (x) = ax + bx, 0 ≤ x ≤ 1


ax − b,
x>1
5. (12 points) Consider the ellipse x2 + xy + y 2 = 3.
(a) Find y 0 by implicit differentiation.
(b) Find the coordinates of the two points on the curve where x = 1.
(c) Find the equations of the normal lines at the two points.
6. (12 points) The graph at right shows the velocity v = s0 (t)
of a particle moving on a coordinate line, 0 ≤ t ≤ g. No
explanation is necessary for the following questions.
(a) When does the particle move forward?
(b) When is the particle at rest?
(c) When is the acceleration of the particle positive?
(d) When does the particle move at its greatest speed?
v á s¢ HtL
v
a
b
c
d
e f g
t
7. (12 points) Let f (x) =
1 − 3x
.
3x2 + 1
(a) Find the domain of f .
(b) Find the absolute maximum and minimum values of f on [−1, 1].
8. (20 points) Sketch an example of each of the functions r, s, u, and v, satisfying the following
conditions. No explanation is necessary.
(a) Function r is odd and lim r(x) = −∞.
x→2
(b) Function s is even, non-linear, and lim s(x) = 2.
x→∞
(c) Function u has a critical number at x = 2 and no local maximum or minimum values.
v(2 + h) − v(2)
(d) Function v has lim
= −2 and v 0 > 0 on (−∞, 1).
h→0
h
1
.
9. (10 points) Let g(x) = √
1+x
(a) Find L(x), the linearization of g, at x = 0.
(b) Use the linearization to estimate the value √
1
.
0.96
10. (8 points) At 9:00 a.m. a car’s speedometer reads 20 mph. At 9:15 a.m. it reads 55 mph. Use the
Mean Value Theorem to show that at some time between 9:00 and 9:15 the acceleration is exactly
140 mi/hr2 .
11. (14 points) Libby and Ralph have arranged to meet in front
of Grusin Music Hall to attend a Takács Quartet recital at 4
p.m. At 3:55 p.m., Libby is 330 yards west of Grusin and
walking east at a constant rate of 90 yd/min. Three minutes
later, Ralph is 80 yards north of Grusin and walking south
toward the music hall. At this moment the distance between
Libby and Ralph is decreasing at a rate of 150 yd/min.
Ralph
Libby
(a) How fast is Ralph walking then?
(b) If Libby and Ralph maintain their speeds, will they
both arrive before the recital begins?
Grusin