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APPM 1350 Midterm 2 Fall 2011

On the front of your bluebook, please write: a grading key, your name, student ID, section, and instructor’s name (Chang, Curry, Dougherty, Guinn, Nelson). This exam is worth 100 points and has 7 questions. Show all work! Answers with no justification will receive no points. Please begin each problem on a new page. No notes, calculators, or electronic devices are permitted.

1. (14 points) Find dy/dx for the following functions. You need not simplify your answers.

(a) y = 3x √

x + √

x (b) y = (

cos x sin x− 1

)2/3

2. (12 points) Let y2 = x3(2− x).

(a) Use implicit differentiation to find dy/dx. (b) Find the linearization at the point (1, 1) and use it to approximate the y value of the function when x = 1.1. (c) If y′(1) > 0 and y′′(1) < 0, can you determine whether the approximation underestimates or overestimates

the true y value on the curve? Explain your reasoning.

3. (10 points)

(a) State the Mean Value Theorem. (b) Use the Mean Value Theorem to find the smallest and largest values that f(3) can have when f(−1) = 2

and 2 ≤ f ′(x) ≤ 4. Explain your reasoning.

4. (8 points) For each of the following, clearly write TRUE or FALSE (not just T or F). No justification is necessary.

(a) If f ′(a) = 0, then f(x) has a local extremum at x = a. (b) If f(x) has a local extremum at x = a, then f ′(a) = 0. (c) If f ′(x) = g′(x) for all x, then f(x) = g(x) for all x.

(d) If f is differentiable, then d

dx f( √

x) = f ′(x) 2 √

x .

5. (25 points) Consider the function f(x) = −x2 + 6x− 10

x− 3 .

(a) Does f have any horizontal, vertical or slant asymptotes? Use appropriate limits to justify your answer.

(b) The first derivative of f is f ′(x) = −x2 + 6x− 8

(x− 3)2 . On what intervals is f increasing? decreasing?

(c) Find all local maximum and minimum values of f .

(d) The second derivative of f is f ′′(x) = −2

(x− 3)3 . On what intervals is f concave up? concave down?

(e) Find all inflection points of f . (f) Sketch a graph of f . Clearly label any intercepts, asymptotes, local extrema, and inflection points.

6. (15 points) A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

at a rate of 1 m!s, how fast is the boat approaching the dock when it is 8 m from the dock?

**19. **At noon, ship A is 100 km west of ship B. Ship A is sailing
south at 35 km!h and ship B is sailing north at 25 km!h.
How fast is the distance between the ships changing at
4:00 PM?

**20. **A particle is moving along the curve . As the par-
ticle passes through the point , its -coordinate
increases at a rate of . How fast is the distance
from the particle to the origin changing at this instant?

**21. **Two carts, A and B, are connected by a rope 39 ft long that
passes over a pulley . The point is on the floor 12 ft
directly beneath and between the carts. Cart A is being
pulled away from at a speed of 2 ft!s. How fast is cart B
moving toward at the instant when cart A is 5 ft from ?

**22. **Water is leaking out of an inverted conical tank at a rate
of 10,000 cm !min at the same time that water is being
pumped into the tank at a constant rate. The tank has height
6 m and the diameter at the top is 4 m. If the water level is
rising at a rate of 20 cm!min when the height of the water
is 2 m, find the rate at which water is being pumped into the
tank.

**23. **A trough is 10 ft long and its ends have the shape of isos-
celes triangles that are 3 ft across at the top and have a
height of 1 ft. If the trough is being filled with water at a
rate of 12 ft !min, how fast is the water level rising when
the water is 6 inches deep?

**24. **A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the
shallow end, and 9 ft deep at its deepest point. A cross-
section is shown in the figure. If the pool is being filled at a
rate of 0.8 , how fast is the water level rising when
the depth at the deepest point is 5 ft?

Gravel is being dumped from a conveyor belt at a rate of 30 , and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are

ft3!min
**25.
**

3 6

12 6166

ft3!min

3

3

A B

*Q
*

*P
*

12 f t

*QQ
Q
*

*P
QP
*

3 cm!s
*x*"4, 2#

*y *! s*x
*

always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

**26. **A kite 100 ft above the ground moves horizontally at a
speed of 8 ft!s. At what rate is the angle between the string
and the horizontal decreasing when 200 ft of string has been
let out?

**27. **Two sides of a triangle are 4 m and 5 m in length and the
angle between them is increasing at a rate of 0.06 rad!s.
Find the rate at which the area of the triangle is increasing
when the angle between the sides of fixed length is .

**28. **Two sides of a triangle have lengths 12 m and 15 m. The
angle between them is increasing at a rate of . How
fast is the length of the third side increasing when the angle
between the sides of fixed length is 60 ?

Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure and volume sat- isfy the equation , where is a constant. Suppose that at a certain instant the volume is 600 cm , the pres- sure is 150 kPa, and the pressure is increasing at a rate of 20 kPa!min. At what rate is the volume decreasing at this instant?

**30. **When air expands adiabatically (without gaining or losing
heat), its pressure and volume are related by the equa-
tion , where is a constant. Suppose that at a
certain instant the volume is 400 cm and the pressure is
80 kPa and is decreasing at a rate of 10 kPa!min. At what
rate is the volume increasing at this instant?

**31. **If two resistors with resistances and are connected in
parallel, as in the figure, then the total resistance , mea-
sured in ohms ( ), is given by

If and are increasing at rates of and , respectively, how fast is changing when and

?

R¡ R™

*R*2 ! 100 !
*R*1 ! 80 !*R
*

0.2 !!s0.3 !!s*R*2*R*1

1
*R
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1
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1
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!
*R
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*R*2*R*1

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***29.
**

#

2#!min

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**132 **! **CHAPTER 2 **DERIVATIVES

7. (16 points) A fish hatchery is constructing a rectangular pool that will be twice as long as it is wide and will hold 9000 cubic feet of water. The reinforced bottom of the pool will cost twice as much per square foot as the sides of the pool. Find the pool dimensions that will minimize the cost of construction.