review problems for exam 2 ce math 1050
TRANSCRIPT
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Review problems for
Exam 2 โ CE Math 1050
This is a comprehensive set of review problems for Exam 2, CE Math 1050. The layout of the document is:
Learning objectives are numbered and bolded. Sample problems for each learning objective are beneath the
objective. Complete answers begin on page 8. For classes taking their exams as written by the UVU Math
Department, the question types (not exact questions) for Exam 2 will be taken from a much smaller subset of
this set.
There is a new request on these problems, that some points be labeled/located on the graph. This means that
either the points are labeled with their coordinates, or the points are clearly marked on the graph at obvious
intersections of the graphing lines to make the location clear. For the answer key, we have chosen to label the
points to make their positions clear.
This exam will โcoverโ sections 3.2 to 4.3 from the Stewart text.
Chapter 3: Polynomial and Rational Functions
A student is able to:
2. Find the vertex and max/min values of a quadratic function.
I. For the following functions: graph the function, indicate the location of the vertex, include the y-intercept
and at least 3 other points well labeled/located on your graph. Then state the range of the function.
a) 242 xxf
b) 562 xxxf
II. A ball is launched upward off a 200-foot cliff and then falls into the ocean below. The formula for height, in feet, t seconds after the ball is thrown is given by
โ(๐ก) = โ16๐ก2 + 64๐ก + 200. a) How many seconds after the ball is tossed does the ball reach its maximum height? [work expected] b) What is the maximum height the ball reaches? [work expected]
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3. Graph a polynomial, showing x- and y-intercepts and proper end behavior.
I. Sketch the graph of the polynomial ๐(๐ฅ) = 1
2(๐ฅ + 3)(๐ฅ โ 2)2(๐ฅ โ 4)3.
What is the leading term? ___________ Describe the end behavior. ____________________________
____________________________ What is/are the x-intercept(s)? _________________________ What is/are the y-intercept(s)? _________________________
What are the zeros of the function and state their multiplicities? ____________________________________ Sketch:
II. Sketch the graph of the polynomial ๐(๐ฅ) = โ๐ฅ3 + 4๐ฅ2 โ ๐ฅ โ 6 What is the leading term? __________ Describe the end behavior. ___________________________ ___________________________ What is/are the x-intercept(s)? ________________________ What is/are the y-intercept(s)? ________________________ What are the zeros of the function and state their multiplicities? ___________________________________ Sketch:
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4. Perform long division on polynomials. (Also done in #13 below)
Divide p(x) by d(x) using long division.
a) p(x) = 2x3 โ 5x2 + 1; d(x) = x2 โ 3 b) p(x) = 10x3 + x2 + 18x + 2; d(x) = 2x2 โ x + 4
5. Perform synthetic division on polynomials. (Used in many objectives below.)
6. Use the Remainder Theorem to evaluate a polynomial.
Let ๐(๐ฅ) = 2๐ฅ4 โ 3๐ฅ3 โ 5๐ฅ2 + ๐ฅ โ 6. Use synthetic division or the Remainder Theorem to evaluate P(-2).
7. Use the Factor Theorem to find a factor of a polynomial. (Used in #11 below, and for other
objectives.)
8. Construct a polynomial given the zeros and their multiplicities.
I. Write a formula for a polynomial which has degree 5; zeros 3, 2i, and -2 (with multiplicity 2). (For this problem, the polynomial need not be multiplied out.)
II. Find a polynomial in standard form with the given conditions. (Remember that Standard form is multiplied
out)
a) degree 3, zeros -2, 0, 3 b) degree 3, zeros 0 (multiplicity 2) and 1
III. Find the unique polynomial in standard form with the given condition.
a) degree 2, zeros -3 and -1, leading coefficient an = โ3
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9. Use the Rational Zeros Theorem, Descartesโ Rule of Signs, and the Upper and Lower Bounds
Theorem in finding zeros of polynomials.
I. Show that 4 is an upper bound of the real zeros of ๐(๐ฅ) = ๐ฅ3 โ ๐ฅ2 + 4. Please, also explain what you see that indicates this is an upper bound.
II. Show that -4 is a lower bound of the real zeros of ๐(๐ฅ) = ๐ฅ3 โ ๐ฅ2 + 4. Please, also explain what you see
that indicates this is a lower bound.
III. For the following function - Use the Rational Zero Theorem to determine all possible rational zeros - Use Descartesโs rule of signs to determine the number of positive and negative zeros - Factor the polynomial - Sketch the graph, showing intercepts and proper end behavior
P(x) = 6x3 + 17x2 + x โ 10
IV. For the following function - Use the Rational Zero Theorem to determine all possible rational zeros (p/qโs) - Use Descartesโs rule of signs to determine the number of positive and negative zeros
a) P(x) = 2x6 - x5 - 13x4 + 13x3 + 19x2 - 32x + 12
b) ๐(๐ฅ) = 4๐ฅ7 โ ๐ฅ5 + 5๐ฅ4 + ๐ฅ3 + 5.
10. Solve polynomial equations.
Solve the polynomial equation: 6x3 + x2 โ 5x โ 2 = 0
11. Factor a polynomial into linear and/or irreducible quadratic factors.
Factor the polynomial into linear factors and/or irreducible quadratic factors.
a) p(x) = x3 โ 3x2 + 4x โ 2
b) p(x) = x3 + 5x2 + 17x + 13
c) p(x) = x4 + 4x3 โ 3x2 โ 10x + 8
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12. Factor a polynomial completely into linear factors real and complex.
I. Given a zero of the polynomial, factor the polynomial into linear factors: p(x) = x4 โ 6x3 + 14x2 โ 24x + 40; ๐ฅ = 2๐ is a zero.
II. Factor the following polynomial completely into linear factors: p(x) = 27x3 โ 8
13. Find vertical and horizontal asymptotes of rational functions.
Find all the asymptotes (vertical and horizontal) of the following rational functions.
a) ๐(๐ฅ) =๐ฅ2โ4
๐ฅ3โ2๐ฅ2โ3๐ฅ
b) ๐(๐ฅ) =(2๐ฅโ3)(๐ฅ+1)
๐ฅ(๐ฅโ3)
14. Graph a rational function, showing intercepts and asymptotes.
I. Consider the rational function ๐(๐ฅ) =๐ฅ+2
๐ฅ2โ4๐ฅ+3.
a) Show and label the vertical asymptote(s), if any.
b) Show and label the horizontal asymptote(s), if any.
c) Show and label the x and y intercepts, if any.
d) Write the domain in interval notation.
e) Graph the function.
II. Graph the rational function ๐(๐ฅ) =๐ฅ2+๐ฅโ2
๐ฅ3โ๐ฅ2โ6๐ฅ. Show and label all asymptotes, intercepts, and hole(s), if any.
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Chapter 4: Exponential and Logarithmic Functions
A student is able to:
1. Graph exponential and logarithmic equations.
I. Starting with the basic function f (x) = log2x , use transformations, as needed, to answer the following
questions for the function )4(log1)( 2 xxg .
a) Identify the y-intercept of g(x) , if any:
b) State the equation of the asymptote of g(x) , if any:
c) Find the domain and the range of g(x)using
interval notation.
d) Sketch the graphs of ๐(๐ฅ) and g(x) . (A complete
graph includes at least 3 labeled/located points on the graph.)
II. Starting with the basic function: f (x) = 2x , use transformations, as needed, to answer the following
questions for the function g(x) = 2x+2 -1.
a) Find the y-intercept of g(x) , if any:
b) Find the asymptote of g(x) , if any:
c) Sketch the graph of g(x) . (A complete graph includes
at least 3 labeled/located points on the graph.)
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2. Solve interest problems using the compound interest formula:
A P 1r
n
nt
and the continuous
interest formula: A = Pert. I. If you put $3200 in a savings account that pays 2% a year compounded quarterly, how much will you have in
the account in 15 years? Use ๐ด = ๐ (1 +๐
๐)
๐๐ก or A = Pert whichever is appropriate. (Please, leave your
answer in a form ready to put into a calculator or you may enter it in your calculator OR round your answer to 2 decimal places.)
II. I want to have $50,000 in 7 years. How much should I invest now in an account earning 6% compounded
continuously? Use ๐ด = ๐ (1 +๐
๐)
๐๐ก or A = Pert whichever is appropriate. (Please, leave your answer in a
form ready to put into a calculator OR enter it in your calculator and round your answer to 2 decimal places.)
3. Switch between exponential and logarithmic forms using the definition of logarithm: ๐ = ๐๐ โ๐๐๐๐๐ = ๐.
I. Write log813 =1
4 in its equivalent exponential form.
II. Write ex = 6 in its equivalent logarithmic form.
III. Evaluate/Simplify each expression using the definition of logarithm
a) ln e3 = ___________________
b) 7log73โ2 = _________________
c) log2โ8 = _________________
d) log2 0.25 = ________________
IV. Solve for x:
a) log2 ๐ฅ = โ6
b) log๐ฅ 81 = 4
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ANSWERS Chapter 3: Polynomial and Rational Functions
2. Problem I:
a) Vertex (4,2); y-intercept (0,18); range [2, โ)
b) Vertex (-3, 4); y-intercept (0, -5); range (-โ, 4]
Problem II:
a) 2 seconds b) 264 feet
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3. Problem I:
The leading term is 1
2๐ฅ6
End behavior: y โ โ as x โ โ; y โ โ as x โ -โ, (The graph goes up on both ends)
x-intercepts: (-3, 0), (2, 0), (4, 0) y-intercept: (0, -384) zeros: x = -3 (mult 1), x = 2 (mult 2), x = 4 (mult 3)
Problem II: The leading term is โ๐ฅ3 End behavior: y โ -โ as x โ โ; y โ โ as x โ -โ,
(The graph goes up on the left and down on the right) x-intercepts: (-1, 0), (2, 0), (3, 0)
y-intercept: (0, -6) zeros: x = -1 (mult 1), x = 2 (mult 1), x = 3 (mult 1)
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4.
a) The quotient is 2๐ฅ โ 5 and the remainder is 6๐ฅ โ 14 OR ๐(๐ฅ)
๐(๐ฅ)= 2๐ฅ โ 5 +
6๐ฅโ14
๐ฅ2โ3
b) The quotient is 5๐ฅ + 3 and the remainder is ๐ฅ โ 10 OR ๐(๐ฅ)
๐(๐ฅ)= 5๐ฅ + 3 +
๐ฅโ10
2๐ฅ2โ๐ฅ+4
5. No questions 6.
-2 2 -3 -5 1 -6
-4 14 -18 34
2 -7 9 -17 28 So P(-2) = 28 7. No questions 8.
Problem I: p(x) = (x โ 3)(x โ 2i)(x + 2i)(x + 2)2 OR if you insist on multiplying it out P(x) = x5 + x4 โ 4x3 โ 8x2 โ 32x โ 48. (Other answers are correct if they are a just a real number multiple
of P(x).) Problem II:
a) ๐(๐ฅ) = ๐ฅ3 โ ๐ฅ2 โ 6๐ฅ, (Other answers are correct if they are a just a real number multiple of P(x).) b) ๐(๐ฅ) = ๐ฅ3 โ ๐ฅ2, (Other answers are correct if they are a just a real number multiple of P(x).)
Problem III: a) P(x) = -3x2 โ 12x โ 9
9.
Problem I: All the numbers in the last row (1, 3, 12, 52) are non-negative so 4 is an upper bound for the real zeros of P.
4 1 -1 0 4
4 12 48
1 3 12 52
Problem II: The numbers in the last row (1, -5, 20, -65) alternate between nonpositive and nonnegative so -4 is a lower bound for the real zeros of P.
-4 1 -1 0 4
-4 20 -80
1 -5 20 -76
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Problem III: Possible rational zeros (p/qโs): ยฑ1, ยฑ2, ยฑ5, ยฑ10, ยฑ1/2, ยฑ1/3, ยฑ1/6, ยฑ2/3, ยฑ5/2, ยฑ5/3, ยฑ5/6, ยฑ10/3 Positive real zeros: one ๐(โ๐ฅ) = โ6๐ฅ3 + 17๐ฅ2 โ ๐ฅ โ 10 So Negative real zeros: two or zero P(x) = (2x + 5)(x + 1)(3x โ 2)
Problem IV: a)
Possible rational zeros: ยฑ1, ยฑ2, ยฑ3, ยฑ4, ยฑ6, ยฑ12, ยฑ1/2, ยฑ3/2 Positive real zeros: four or two or zero
๐(โ๐ฅ) = 2๐ฅ6 + ๐ฅ5 โ 13๐ฅ4 + 19๐ฅ2 + 32๐ฅ + 12 So Negative real zeros: two or zero
b) Possible rational zeros: ยฑ1, ยฑ5, ยฑ1/2, ยฑ5/2, ยฑ1/4, ยฑ5/4 Positive real zeros: two or zero
๐(โ๐ฅ) = โ4๐ฅ7 + ๐ฅ5 + 5๐ฅ4 โ ๐ฅ3 + 5 So Negative real zeros: three or one 10. x = 1, x = -1/2, x = -2/3 11. a) p(x) = (x โ 1)(x2 โ 2x + 2)
b) p(x) = (x + 1)(x2 + 4x + 13)
c) p(x) = (x โ 1)2(x + 2)(x + 4)
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12. Problem I: p(x) = (x + 2i)(x โ 2i)(x โ (3 โ i))(x โ (3 + i))
Problem II: ๐(๐ฅ) = (3๐ฅ โ 2) (๐ฅ โ (โ1
3โ
๐โ3
3)) (๐ฅ โ (โ
1
3+
๐โ3
3))
= (3๐ฅ โ 2) (๐ฅ +1
3+
๐โ3
3) (๐ฅ +
1
3โ
๐โ3
3) (Since the first term of the polynomial
we were factoring was 27๐ฅ3, and ours would only be 3๐ฅ3, we must multiply our answer by 9. We will multiply each of the last two factors by 3, which also serves to get
rid of the fractions, too.)
= (3๐ฅ โ 2)(3๐ฅ + 1 + ๐โ3)(3๐ฅ + 1 โ ๐โ3)
13. a) vert. asymptotes: x = 0, x = -1, x = 3 , horiz. Asymptote: y = 0 b) vert. asymptotes at x = 0 and x = 3, horizontal asymptote at y = 2 14. Problem I: Vertical asymptotes at x = 1 and x = 3 Horizontal asymptote at y = 0 x-intercept at (-2, 0) y-intercept at (0, 2/3) Domain (-โ, 1) u (1, 3) u (3, โ)
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Problem II: Vertical asymptotes at x = 0 and x = 3 Horizontal asymptote at y = 0 x-intercept at (1, 0) No y-intercept Hole at (-2, -3/10)
Chapter 4: Exponential and Logarithmic Functions
1. Problem I:
a) y-intercept: (0,3) b) Vertical asymptote at x = -4 c) Domain (-4, โ), Range (-โ, โ) d) Sketch:
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Problem II:
a) y-intercept: (0,3) b) Horizontal Asymptote at y = -1
c) Sketch (the graph of f (x) = 2x need not be present for full credit.)
2.
Problem I: At the end of 15 years, there will be 3,200 (1 +0.02
4)
(4ร15) ๐๐ 3,200(1.005)60 dollars in the
account OR approximately $4,316.32
Problem II: We will need to invest 50,000
๐(0.06ร7) ๐๐ 50,000
๐(0.42) dollars initially OR approximately $32,852.34
3.
Problem I: 81(1 4โ ) = 3 Problem II: ๐๐๐๐6 = ๐ฅ OR ln6 = x Problem III:
a) 3
b) ๐โ๐ ๐๐ ๐
๐
c) log2 โ8 = log2 23
2 = ๐
๐
d) log2. 25 = log21
4= log2 2โ2 = โ๐
Problem IV:
a) ๐ฅ =1
64
b) ๐ฅ = 3