Test 2 Review

TEST 2REVIEW

NOTE: You should also go over the Test 1 Review Sheet and the Additional Problems in the Unit 2 Homework. Remember, the tests are cumulative and you will also be tested on the Unit 1 material.

PART 1 – COMPUTER SECTION

Directions:You may use computer software or a graphing calculator on this part of the review. However, complete solutions must be given with clear explanations.

A.Text:Chapter 3 Review Exercises page 248: 41, 46 (tangent line only), 47.

Chapter 1 ReviewExercises page 82: 31a.

Exercises page 205: 85.

  1. Additional Problems:
  1. a. At what point(s) does the graph of have a tangent line parallel to ?

b. Write the equation(s) of the tangent line(s).

c. Verify your answer with a graph.

  1. Consider the curve defined by the parametric equations:

for .

a. Obtain a graph of the curve. Copy the curve onto graph paper, label the tic marks on each axis, and use arrows to show the direction of the plot. (Use Converge and freeze the graph at the first point.) Be sure you have a complete graph.

b. Use algebra to find the x–intercepts exactly. Show the equations that you solved.

c. Find the equation of the tangent line at the point (3, 0) and add its graph to part a.

PART 2 – NON COMPUTER SECTION

Directions: You may not use computer software or a graphing calculator on these questions.

A.Text:Chapter 3 Review, page 248:

Concept Check: 2.

True-False Quiz: 1 – 9 all, 12.

Exercises: 1 – 16 all, 18, 19, 20 (use logarithmic differentiation), 25, 27, 28, 29, 37, 38, 64, 68, 73.

Chapter 2 Review Exercises,page 165: 33.

Chapter 4 Review Exercises, page 324: 36, 37.

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  1. Additional Problems:

Note: Related rate problems could be on Part 1 - Computer and/or Part 2 - Non computer.

1.A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is (a) 7 feet, (b) 15 feet, and (c) 24 feet from the wall?

2.Gravel is being dumped from a conveyor belt at a rate of 10 ft/min, and its coarseness is such that it forms a pile in the shape of a cone. The diameter of the base of the cone is approximately three times the height. How fast is the height of the pile increasing when the pile is 15 feet high?

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test 2 review solutions

Test 2 Review Solutions

test 2 review solutions

Part 1

A.

Chapter 3

46.Equation of the tangent line is

B.

1.

The tangent line to has a slope at each point that is given by its derivative,

. If the tangent line is parallel to , then the slope of the tangent line must be 6, so or .

a. The two points of tangency are (–2, 5) and

(1, 9.5).

b. The equations of the two tangent lines

are:

c.

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2. a.

  1. To find the x-intercepts, let y=0.

in [0, 2]

.

The x-intercepts are given in this table:

t / x
0 / 1
/ –3
 / –1
/ 3
2 / 1

c.At (3, 0), from above table, t =

At (3, 0), the tangent line is

Part 2

A.

Chapter 3 Review

Concept Check

2.See Reference page 5 in the back of your text.

True False

2.False. Product Rule.

4.True by the Chain Rule.

6.False. is a constant so its derivative is 0.

8.False. is a constant so its derivative is 0.

12.False. A tangent line to the parabola has slope , so at (−2, 4) the slope of the tangent line is 2(−2) = −4 and the equation of the tangent line is . The equation given is not even linear.

Exercises

2.

4.

6.

8.

10.Hint: m and n are constants.

12.

14.

16.

18.

20.

28.

38.

64.

68. a.

b. when t > 2 , so it moves

upward when t >2, downward when

.

c. Distance upward:

Distance downward:

Total distance:

d.

e. The particle is speeding up when v and a

have the same sign ( t > 2). The particle is

slowing down when v and a have opposite

signs; that is, when 0 < t < 2.

Chapter 4 Review

36. The water level is rising at a rate of centimeters per second when h = 5.

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B.

1.

Given: The ladder is 25 feet long.

ft/sec

Find: when x = 7 ft and when x = 15 ft and when x = 24 ft.

From the Pythagorean Theorem, .

Differentiating both sides with respect to time,

Or

Solving for , we get

Substituting for ,

a. When x = 7, y = 24 and ft/sec

Note that the rate is negative because the length (y) is decreasing.

b. When x = 15, y = 20 and ft/sec

c. When x = 24, y = 7 and ft/sec

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2.

Given: and

Find: when h = 15 ft

Volume of a cone = ;

d = 2r = 3h

So,

When h = 15 ft and ,