Download Math 104 - Precalculus Midterm Fall 2012 Name KUID Instructor Part 1

Math 104 - Precalculus
Midterm
Fall 2012
Name
KUID
Instructor
Part 1: Multiple Choice (100 points). Show work in the space provided. Circle the correct
answer. Partial credit will be given only if work is shown. Each problem is 10 points.
1. Evaluate the difference quotient
f (x + h) − f (x)
h
for the function f (x) = 3x2 + x − 1.
(a) 6x + 3h + 1
(b) 6x + 1
(c) 3x + h + 1
(d) 3x + 3h + 1
(e) 1
2. A manufacturer determines if they sell x units in one day, their revenue (in dollars) is given
by R(x) = −.002x2 + 8x + 2200 and that the total cost (in dollars) of selling x units in one
day is given by C(x) = −.001x2 + 5x + 4000. What is their maximum profit per day?
(a) $ 10200
(b) $ 2000
(c) $ 1500
(d) $ 450
3. The average rate of change of the function y = f (x) between x = a and x = b is :
(a) The slope of the secant line between x = a and x = b on the graph of f
f (a) + f (b)
(b)
2
(c) The instantaneous rate of change of f at the midpoint between (a, f (a)) and (b, f (b))
b−a
(d)
f (b) − f (a)
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Math 104 - Precalculus
Midterm
Fall 2012
4. Given the graph of f (x) , circle the letter below the graph of − 12 f (x)
3
2
1 f (x)
−4 −3 −2 −1
−1
1 2 3 4
3
2
1
−4 −3 −2 −1
−1
−2
−3
3
2
1
1 2 3 4
−2
−3
−4 −3 −2 −1
−1
−2
−3
f (x)
(A)
(B)
3
2
1
3
2
1
3
2
1
−4 −3 −2 −1
−1
1 2 3 4
−4 −3 −2 −1
−1
−2
−3
1 2 3 4
−2
−3
(C)
1 2 3 4
(D)
−4 −3 −2 −1
−1
1 2 3 4
−2
−3
(E)
5. If f (3) = 2, f (4) = 3, g(2) = 5, and g(3) = 2, and h = (g ◦ f −1 )(x), then h(2) =
(a) 2
(b) 3
(c) 4
(d) 5
6. Find the inverse of the function j(x) = log3 (x − 1) + 4
(a) j −1 (x) = − log3 (x + 1) − 4
(b) j −1 (x) = 3x+4 − 1
(c) j −1 (x) = 3x−4 + 1
(d) j −1 (x) = (x − 4)3 + 1
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Math 104 - Precalculus
7. Expanding
(x − 1)4 y 3
ln( √
)
z(1 + x)
Midterm
Fall 2012
using the properties of logarithms, we get
(a) 4 ln(x − 1) − 3 ln y + 21 ln z + ln(1 + x)
(b) 4 ln(x − 1) + 3 ln y + 12 ln z + ln(1 + x)
(c) 4 ln(x − 1) + 3 ln y − 12 ln z − ln(1 + x)
(d) ln(4(x − 1) + 3y − 12 z − (1 + x))
8. List all of the asymptotes of the function g(x) =
2x2 − x
x2 − 9
(a) Vertical: x = 3, x = −3; Horizontal: none
(b) Vertical: none; Horizontal: y = 0
(c) Vertical: x = 9; Horizontal: y = 2
(d) Vertical: x = 3, x = −3; Horizontal: y = 2
9. What is 10 + 3i multiplied by its conjugate equal to?
(a) 91 − 60i
(b) 91
(c) 109
(d) 100 + 30i
10. Suppose P (x) is a polynomial. Which statement is not equivalent to the others?
(a) (−1, 0) is an x-intercept of the graph of P
(b) (x − 1) is a factor of P
(c) x = −1 is a zero of P
(d) P (−1) = 0
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Math 104 - Precalculus
Midterm
Fall 2012
Part 2: Long answer (100 points). Show all your work and steps. Each question will be
graded based on the accuracy of work shown. Answers should be exact unless otherwise
stated. Each problem is worth 20 points.
11. Consider the functions
√
f (x) = − x − 1 − 1,
g(x) = 3x−1 + 1
(a) Evaluate (f ◦ g)(3)
(b) Identify the underlying basic function of f . List in order the transformations of f from
the basic function.
(c) Determine the domain and range of f (you may determine range graphically).
(d) Sketch a graph of f , labeling 3 points.
(e) Find f −1 (x) algebraically.
(f) State the domain and range of f −1 .
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Math 104 - Precalculus
Midterm
Fall 2012
12. Consider the polynomial p(x) = x4 − 6x3 + 14x2 − 16x + 8.
(a) List all possible rational zeros of p(x).
(b) Factor the polynomial completely into linear factors (you must show all work).
(c) State all the complex zeros of p, and state the multiplicity of each zero.
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Math 104 - Precalculus
Midterm
13. Algebraically solve exactly three (3) of the below equations for x.
Circle the letters of the three you choose to be graded and show all work.
(a) log2 (x2 − 6x) − 6 = −2
(b) e3x e5 = (ex )2
(c) log4 (x + 3) = 2 − log4 (x − 3)
(d) e2x − 6ex = −8
(e) ln(x + 3) − ln x = ln 6
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Fall 2012
Math 104 - Precalculus
Midterm
14. Assume and use an Exponential Decay model. (Leave answers in exact form.)
A sample of bismuth-210 decayed to 33% its original mass after 8 days
(a) Find the decay constant r, for the Exponential Decay model.
(b) Find the half-life of this element, bismuth-210.
(c) Find the percentage of the original mass remaining after 12 days.
(approximate using 2 decimals)
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Fall 2012
Math 104 - Precalculus
Midterm
Fall 2012
15. Solve the inequality algebraically, use a sign-test table (chart or number line with test points).
State the solution in interval notation.
1+x
≤1
1−x
Bonus Answer any of the following questions, they are worth 7, 6 and 7 points respectively.
(a) Find the equation of the circle with endpoints of a diameter at P (−1, 1) and Q(5, 9).
(b) Find the equation of the line which passes through points P and Q.
(c) Find the equation of the line which is perpendicular to the line in (b), and passes through
the center of the circle found in (a).
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