test 3 review
TEST 3 REVIEW
NOTE: You should also go over the Test 1 and Test 2 Review Sheets and the Additional Problems in the Unit 2 and Unit 3 Homework. Remember, the tests are cumulative and you will also be tested on the Unit 1 and Unit 2 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 4 ReviewExercises page 324: 1 – 5, 7 – 10, 14, 15, 17, 19, 25, 37.
Chapter 5 Review Exercises page 425: 2 (Use Converge for part b.)
- Additional Problems:
- Consider the function
a. Obtain a complete graph of the function.
b. Find the intercepts.
c. Obtain the equations of any asymptotes.
d. Find the critical numbers.
e. Test each critical number to determine whether a local minimum or local maximum occurs there.
f. Find the coordinates of any relative extrema.
g. Find the coordinates of any points of inflection.
h. On what intervals is f(x) concave up or concave down?
2.If a rectangular box with a square base and open top is to have a volume of 4 , find the dimensions that require the least amount of material.
3.A rectangular box with a square base and an open top is constructed from two types of material. The material used to make the bottom of the box costs $0.10 per square inch; the material used to make the rest of the box costs $0.06 per square inch. The total cost of the box is to be $3.00. If x is one of the sides of the base of the box and h is the height, find the value of x that maximizes the volume.
4.A cylindrical can with closed bottom and closed top is to be constructed to have a volume of 1 gallon (approximately 231 cubic inches). The material used to make the bottom and top costs $0.06 per square inch, and the material used to make the curved surface costs $0.03 per square inch. Find the dimensions of the can (radius and height) that minimize the total cost and determine what that minimum cost is.
149
PART 2 – NON COMPUTER SECTION
Directions: You may not use computer software or a graphing calculator on these questions.
- Text:Chapter 2 Review Exercises page 165: 40, 43-45.
Chapter 4 Review page 323:
Concept Check: 1– 6, 8
True-False Quiz: 1 – 7
Exercises: 1-5 odd, 51-56.
Chapter 5 Review page 423:
Concept Check: 1, 2, 6, 7.
True-False Quiz: 1– 4, 6.
Exercises: 1 a, 4, 5, 9 – 21.
- Additional Problems:
For Problems 1 – 15, evaluate the integrals and check your results by differentiating the answers.
test 3 review
test 3 review
1.
2.
3.dx
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
test 3 review
150
For problems 16 – 22, evaluate the definite integrals using the Evaluation Theorem (the second fundamental theorem of calculus).
test 3 review
test 3 review
16.
17.
18.
19.
20.
21.
22.
test 3 review
test 3 review
23.Below are the graphs of three functions. One of these graphs is f, one is the graph of , and one is the graph of . Identify which graph is which and give reasons.
a)c)
b)
151
24.Sketch the graph of a continuous function having the indicated characteristics:
if x < 3
if x > 3
is undefined
25.Sketch the graph of one continuous function f(x) that satisfies all of the stated conditions:
26.The graph of is shown below.
a. If the domain of is all real numbers, sketch a graph of.
b. Find approximate values of all critical numbers of the original function .
Explain clearly why each one is a critical number.
152
test 3 review solutions
Test 3 Review SOLUTIONS
Part 1
test 3 review solutions
A.
Chapter 4 Review
8.a. No asymptote.
b. f is increasing on and decreasing
on
c. Local minimum ; no local
maximum
d.Concave down on (–2,0) and concave up on and. Inflection point at (0,0) and (–2,–16).
e.
10.a. Vertical asymptotes:
Horizontal asymptote:y=0
b. fis increasing on and and
decreasing onand
c. Local minimum ; no local maximum
d. Concave down on and.) and
concave up on . No inflection point
e.
14.a. Vertical asymptotes:
No horizontal asymptote
b.fis increasing on and decreasing on. Domain of f is
c.No local maximum or minimum
d.Concave down on and.).
No inflection point
e.
Chapter 5 Review
2.a.
The Riemann sum represents the sum of the areas of the two rectangles above the x-axis minus the area of the rectangle below the x-axis.
b.Final estimate from Converge: .6669108
c.
153
d.= net area, that is, area above the
x-axis minus the area below the x-axis.
B.
1.a.Graph:
b.x-intercepts: when x = 0.38197 and x = 2.6180; there is no y-intercept.
c.Asymptotes: There are vertical asymptotes at
x = 0 and x = 4. There is also a horizontal asymptote at y = 1.
d.Derivatives:
Critical numbers:
or
e.Second Derivative Test:
, so
is a
relative or local minimum.
154
, so is a relative or local maximum.
f.is a local minimum; is a local maximum.
g.Points of inflection:
Test concavity on either side of each vertical asymptote and each possible inflection point:
Therefore, is an inflection point.
h.The graph is concave down on and concave up on
2.
Least material means minimize surface area:
Volume is 4 ft3:
is the function to be minimized and its domain is
x > 0.
and ft.
Use either the first or second derivative test to verify that this does yield a relative minimum.
Answer: The box should be 2 ft by 2 ft with a height of 1 ft. The surface area will then be 12 square feet.
3.Same diagram as #3.
Maximize the volume:
Cost constraint:
This is the function to be maximized and its domain is x > 0.
in.
Use either the first or second derivative test to verify that this does yield a relative maximum.
Answer: The box should be approximately 3.16 in by 3.16 in with a height of 2.64 in. The volume will then be about 26.4 cubic inches.
4.
Minimize the cost:
The volume is 231 cubic inches:
Then
This is the function to be minimized and its domain is r > 0.
in.
Use either the first or second derivative test to verify that this does yield a relative minimum.
Answer: The can should have a radius of about 2.64 inches and a height of about 10.56 inches. The cost to make the can will be about $7.88.
Part 2 A
Chapter 2 Review Exercises
40.The graph of a has tangent lines with positive slope for x<0 and negative slope for x>0, and the values of c fit this pattern, so c must be the graph of the derivative of a. The graph of c has horizontal tangent lines to the left and right of the x-axis and b has zeros at these points, so b is the graph of the derivative of c. Therefore, a is the graph of f, c is the graph of , and b is the graph of .
44.a.
b.
Chapter 4 Review
Concept Check
2.a. See Theorem 4.2.3 on page 264.
b.See the Closed Interval Method before Example 6 on page 266.
4.See the Mean Value Theorem on page 272. Geometric interpretation: there is some point P on the graph of a function f on the interval (a,b) where the tangent line is parallel to the secant line that connects (a, f(a)) and (b, f(b))
155
6.a. See the First Derivative Test after Example 2
on page 274.
b. See the Second Derivative Test before
Example 4 on page 275.
c.See the note before Example 5 on page 276.
8.Without calculus you could get misleading graphs that fail to show the most interesting features of a function. See the first paragraph on page 282.
True False
2.False. For example, has an absolute minimum at 0, but does not exist.
4.True. By the Mean Value Theorem.
6.False.For example, y=1 has no inflection points but for all c.
Exercises
52.
54.
56.
Chapter 5 Review
Concept Check
2.a. See page 343.
b. See Figure 2 on page 344.
c. See Figures 3 and 4 and the paragraph next to
them on page 344.
6.a. is the family of functions . Any two such functions differ by a constant.
b. The connection is given by the Evaluation
Theorem. See page 356.
True False
2.False. A counterexample is if a = 0, b = 2, and .
4.False. You can’t take a variable outside the integral sign. For example, using on
[ 0,1 ] you get a different answer if you try to take the x outside the integral.
6.True, by the Net Change Theorem, page 360.
156
Exercises
4.
10.
12.
14.
16.
18.
20.
Part 2 B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.Looking at the extrema of each graph and the zeros, it appears that graph (c) is the derivative of graph (a), and graph (b) is the derivative of graph (c). Therefore f is graph (a), is graph (c), and is graph (b).
157
24.One possible graph:
25.One possible graph:
158
26a.
b.Critical numbers at: , x = 0, and because at , and , and.