APPM 1340
Exam 3
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. (10 points)
�y5 � 2 y3 � y2 � x4 � 2 x3 � x2
y
x
Find an equation of the line tangent to the “bouncing wagon” curve shown above at (1, −1).
2. (15 points) Differentiate the following functions. You need not simplify.
2x − 3
− 5x + 1)2
+ cot
(b) s = csc2 3πt
2
(a) y =
(x3
3πt
2
3. (15 points) Functions f (x), g(x), h(x), and s(x) are continuous on the open interval (a, b).
Sketch examples of f, g, h, and s that satisfy the following conditions.
(a) f has no absolute extrema.
(b) g has absolute extrema where g 0 = 0.
(c) h has an absolute maximum at interior point x = c and h0 (c) does not exist.
(d) s is even and has absolute extrema at interior points.
4. (15 points) Find the absolute maximum and minimum values of
f (x) =
on the interval [−1, 2].
x4
− x3 − 2x2
4
5. (15 points)
(a) Find the linear approximation of the function g(x) = √
2
at x = 5.
9−x
(b) Use the approximation to estimate the value of g(5.4).
6. (10 points) A long rope is wrapped around the earth at the equator. A second rope also
encircles the equator but 1 meter off the ground.
(a) Use differentials to calculate the difference in length between the two ropes. (The diameter of the earth is about 12700 km.)
(b) A third rope is 1 meter longer than the first rope. If it also encircles the earth, use
differentials to calculate the distance between the third rope and the ground.
7. (20 points)
It is midnight and the Incredible Hulk is getting angry. His rage causes him to grow in height
at a rate of 0.2 ft/sec. A spotlight on the ground 40 ft in front of him casts a shadow on a wall
10 feet behind him.
(a) How fast is the length of his shadow growing when the Hulk is 7 ft tall?
(b) After reaching his full height of 8 ft the Hulk begins to walk toward the spotlight at
9 ft/sec, intending to destroy it. How fast is the length of his shadow growing when the
Hulk is 10 ft from the spotlight?
Extra Credit (10 points)
Find equations of the two lines tangent to the curve 2x2 + 3y 2 = 5 that pass through the point
(5/2, 0).