Download Exam Final

APPM 1340
Final Exam
Fall 2010
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name,
(2) section number, and (3) a grading table on the front of your bluebook. Start each problem on
a new page. Simplify your answers. A correct answer with incorrect or no supporting work may
receive no credit, while an incorrect answer with relevant work may receive partial credit. Unless
otherwise indicated, show all work.
1. (15 points) Shown below is a graph of the circle x2 + y 2 − 2y = 0.
y
x
(a) Find dy/dx and d2 y/dx2 . You need not simplify.
√
(b) Find an equation of the line tangent to the curve at
3 3
,
.
2 2
2. (20 points) Evaluate the following limits.
2t − 18
=
81 − t2
t→9
√
√
x− 5
(b) lim 2
=
x→5 x − 4x − 5
(a) lim+
(c) lim
θ cot 4θ
=
3
(d) lim
csc(x + h) − csc x
=
h
θ→0
h→0
3. (15 points) Sketch examples of functions f (x), g(x), and h(x) that satisfy the following conditions.
(a) f is odd and f has a local minimum value at x = 2.
(b) g has no local extrema and g 0 (−1) = 0.
(c) h0 (1) does not exist and h00 > 0 for all x 6= 1.
4. (10 points)
y
y � f �x�
2
1
�2
�1
1
2
3
4
5
x
�1
The function f (x) is defined on (−2, 5]. Refer to the graph of f (x) above to answer the
following questions. No explanations are necessary.
(a) For what values of x is f not continuous?
(b) For what values of x is f not differentiable?
(c) Find the following values.
lim f (x) =
x→−2+
lim f (x) =
x→1
lim f (x) =
x→3
√
5. (10 points) Let g(x) = −2x x2 + 3 + 1.
(a) Use the Intermediate Value Theorem to show that the equation g(x) = 0 has at least one
real root.
(b) Find g 0 . You need not simplify.
6. (30 points) Let f (x) =
2
−4 − 6x
−8
, f 0 (x) =
, f 00 (x) =
.
2
1 + 2x
(1 + 2x)
(1 + 2x)3
(a) Find the domain of f .
(b) Does f have any horizontal, vertical, or slant asymptotes?
(c) Find the x and y-intercepts, if any.
(d) On what intervals is f increasing or decreasing?
(e) On what intervals is f concave up or down?
(f) Sketch a graph of y = f (x). Label asymptotes, local extrema, and inflection points, if
any.
7. (10 points) Consider the following function.

2

x + 9
h(x) = 12 4

√3x
(a) Show that h is continuous at x = 3.
(b) Is h differentiable at x = 3? Explain.
x<3
x≥3
8. (15 points) Let f (x) = sec( x3 ) tan( x3 ).
(a) Find f 0 .
(b) Find the linearization of f at x = 3π. Use the linearization to estimate the value of f (9).
√
9. (10 points) The geometric mean of two positive numbers a and b is the number ab. Show
that the value of c in the conclusion
of the Mean Value Theorem for f (x) = 1/x on any
√
positive interval [a, b] is c = ab.
10. (15 points)
A conductor-less runaway train is speeding along a
track at 60 mph, toward a junction 8 miles away. A
single locomotive, unaware of the runaway train, is
heading toward the same junction on a perpendicular track.
(a) If the distance between the trains is 10 miles and decreasing at a rate of 78 mph, find the
speed of the locomotive.
(b) If the trains maintain their speeds, will the locomotive pass the junction before the runaway train arrives?
Extra Credit (15 points)
Two sides of a triangle have lengths 2√
m and 3 m. The angle
between them is increasing at a rate of 7 radians per minute.
How fast is the length of the third side increasing when the
angle between the sides of fixed length is π/3 radians?
(Hint: You may use the Law of Cosines: c2 = a2 + b2 − 2ab cos C.)
A
b
C
c
a
B