Pre-Calculus Home Work

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MATH 115 6381 (2182) Mid Term Exam

Name___________________________________

By signing my name above, I certify that I have completed this assignment individually,

working independently and not consulting anyone except the instructor.

Instructions for Mid-Term Exam:

The Mid-Term Exam consists of 12 problems for a total of 100 points. It is based on the material

covered in the first 3 weeks of the course, but with considerable focus on the Week 4 material:

Chapter 6 (Sections 6.2 through 6.5) of the “College Algebra, 3 rd

Corrected Edition” text and

Chapter 10 (Section 10.1) of the “College Trigonometry, Stitz and Zeager” text. It is open book and

open notes. It is open book and open notes. This means that you may refer to your textbook, notes,

and online classroom materials, but you must work independently and may not consult anyone.

YOU MAY USE YOUR GRAPHING CALCULATOR (TI-83/84 or EQUIVALENT, nothing more

sophisticated) TO ASSIST IN GRAPHING OR ANY NUMERICAL CALCULATIONS).

You must show your work. Answers without any work may earn little, if any, credit. You may type

or write your work in your copy of the exam, or if you prefer, create a document containing your

work. Scanned work is acceptable also.

When you have completed the exam, upload your solutions file to the Mid-Term Exam assignment

in the Assignment folder. DUE DATE is NLT 11:59pm EDT, Sunday, September 16.

If you have any questions, please contact me by e-mail.

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1. (8 points). Find all real solutions to the following equation:

√𝒙 − 𝟐 + √𝒙 − 𝟓 = 4

Answer: x = ______________________

2. (8 points). Consider the following functions:

f(x) = 3 – x 2 ; g(x) = √𝒙 + 𝟏

(a). Compute the composition function (f ◦ g)(x).

Answer: (f ◦ g)(x) = ___________________________ (b). Compute (f ◦ g)(2).

Answer: (f ◦ g)(2) = ________________

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3. (8 points). (a) Rewrite the following exponential expression as a logarithmic expression.

( 𝟏

𝟑 )

−𝟐

= 𝟗

Answer: ___________________________________

(b). Rewrite the following logarithmic expression as an exponential expression:

𝐥𝐨𝐠𝟑 ( 𝟏

𝟖𝟏 ) = −𝟒

Answer: ___________________________________

4. (8 points). Use the properties of logarithms to condense the expression below as a single

logarithm.

𝟏

𝟐 𝐥𝐨𝐠𝟑(𝐱) − 𝟐𝐥𝐨𝐠𝟑(𝐲) − 𝐥𝐨𝐠𝟑(𝐳)

Answer: ___________________________________

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5. (8 points). A can of soda at 83 o F is placed in a refrigerator that maintains a constant

temperature of 35 o F. The temperature T of the soda t minutes after it is placed in the refrigerator

is given by the equation below:

T(t) = 35 + 48 e – 0.058 t

Find the temperature of the soda 25 minutes after it is placed in the refrigerator. (Round to the

nearest degree.)

Answer: ___________________________________

6. (8 points). Solve the following exponential equation for x:

𝟖𝟐𝒙−𝟕 = 𝟔𝟒

Answer: x = ____________________

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7. (10 points). Solve the following logarithmic equation for x (Hint: Use properties of

logarithms to condense the expression on the left-hand side of the given equation FIRST. Also,

check all solutions in the original equation):

𝐥𝐨𝐠𝟓(𝟐𝐱 + 𝟏) + 𝐥𝐨𝐠𝟓(𝐱 + 𝟐) = 𝟏

Answer: x = ________________________

8. (10 points). Solve the rational equation shown below for x. (Check all solutions in the

original equation).

𝟐𝒙 + 𝟏𝟕

𝒙 + 𝟏 = 𝒙 + 𝟓

Answer: x = _______________________________

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9. (14 points). For the quadratic function shown below, answer the following 7 questions:

f(x) = -2(x + 1) 2 + 4

(a). Determine the coordinates of the x-intercepts (if any).

Answer: Coordinates of x-intercepts: ______________________________

(b). Determine the coordinates of the y-intercept.

Answer: Coordinates of y-intercept: ______________________________

(c). Determine the domain of f(x).

Answer: Domain of f(x) (in interval notation): ______________________

(d). Determine the range of f(x).

Answer: Range of f(x) (in interval notation): ______________________

(e). Determine the coordinates of the vertex.

Answer: Coordinates of the vertex: __________________________

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(f). Identify the coordinates of any relative minimum or maximum.

Answer: Relative MINIMUM or MAXIMUM (CIRCLE WHICH IT IS):

Coordinates: _____________

(g). Sketch a graph of f(x) on the coordinate axes below.

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10. (6 points). Suppose we are given the function f(x) = √𝒙.

(a). Find a formula for a function g(x) whose graph is obtained from the above f(x) by shifting

f(x) right 2 units, followed by a shift 3 units down.

Answer: g(x) = ________________________

(b). Find a formula for a function h(x) whose graph is obtained from the above f(x) by shifting

f(x) 1 unit up, followed by a reflection across the x-axis.

Answer: h(x) = _____________________

11. (6 points).

(a). Convert the angle given in degrees to radians. Express your answer in terms of π.

54 o

Answer: 54 o = ____________________ radians

(b). Convert the angle given in radians to degrees.

𝟕

𝟒 𝝅

Answer: 𝟕

𝟒 𝝅 radians = ____________________ degrees

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12. (6 points).

(a). Find a positive angle (measured in degrees) less than 360 o that is co-terminal with the

given angle.

548 o

Answer: ____________________ degrees

(b). Find a positive angle (measured in radians) less than 2π that is co-terminal with the

given angle.

𝟏𝟑

𝟔 𝝅

Answer: ____________________ radians