Mathematics Exam Instructions

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Questions and Answers

What is the price of a cup of coffee 10 years after the cafe opened?

$3.05

$2.25

$2.65 (correct)

$2.50

The linear equation P = 0.04t + 2.25 correctly describes the price of coffee at the cafe.

False

What is the formula for the price of coffee in terms of the number of years since the cafe opened?

P = 0.04t + 2.25

If Alice wants to have $2,000 after 5 years at an interest rate of 3% compounded continuously, which option is her initial investment?

$1694.03 Signup and view all the answers

The function g(x) = e^{2x} represents _______.

an exponential function Signup and view all the answers

Calculate g ◦ f(169). What is the rounded result to the nearest hundredth?

403.43 Signup and view all the answers

The cafe opened with a cup of coffee priced at $3.05.

False Signup and view all the answers

Match the financial terms with their definitions:

Interest Rate = The percentage charged on borrowed money Compounded Continuously = Interest calculated on the initial principal and the accumulated interest Investment = The action of putting money into financial schemes Signup and view all the answers

Which of the following inequalities has the solution set described by the interval notation (−∞, −2] ∪ [0, 1]?

x^3 + x^2 ≥ 2x Signup and view all the answers

The interval notation (−∞, −2) indicates that -2 is included in the solution set.

False Signup and view all the answers

What is the general form of the solution set for the inequality x^3 + x^2 ≥ 2x?

(−∞, −2] ∪ [0, 1] Signup and view all the answers

The range of inputs for the arccos function is from ______.

−1 to 1 Signup and view all the answers

Match the following expressions with the correct descriptions:

arccos(x) = Inverse cosine function x^3 + x^2 = Cubic polynomial (−∞, −2] = Interval including -2 (0, 1) = Open interval between 0 and 1 Signup and view all the answers

What is the correct output of the function arccos(√−3)?

Undefined Signup and view all the answers

The cubic polynomial x^3 + x^2 can be factored as x^2(x + 1).

True Signup and view all the answers

What are the critical points of the inequality x^3 + x^2 - 2x ≥ 0?

−2, 0, 1 Signup and view all the answers

What is the correct equation represented by option (A)?

y = (−2/5)x + 7/5 Signup and view all the answers

Which of the following represents the union of intervals excluding 0 and -2?

(−∞, −2) ∪ (−2, 0) ∪ (0, 2) ∪ (2, ∞) Signup and view all the answers

The degree of a polynomial affects the number of roots it can have.

True Signup and view all the answers

Evaluate the difference quotient for f(x) = 3x^2 + 1.

6x + 3h Signup and view all the answers

The function f(x) = A cos(Bx) can have A as a negative value.

False Signup and view all the answers

What is the perimeter function P(x) for the rectangle formed by a point in the first quadrant on the line y = 6 − 2x?

P(x) = 12 − 2x Signup and view all the answers

In a right triangle, sec θ is the reciprocal of the ______.

cosine Signup and view all the answers

What are the values of A and B if the function is represented as f(x) = 3 cos(2x)?

A = 3, B = 2 Signup and view all the answers

The function P(x) = 12x − 6 represents the perimeter of the rectangle formed by a point (x, y) on the line y = 6 − 2x.

False Signup and view all the answers

If cos θ = 3/5 for θ in Quadrant IV, what is the value of tan θ?

-4/3 Signup and view all the answers

Which of the following statements about the degree and leading coefficient of a polynomial is true?

The degree of the polynomial is 5 and the leading coefficient is negative. Signup and view all the answers

The asymptotes for the function y = (3x − 3)/(x^2 − 8x + 16) are ______.

y = 0, x = 4 Signup and view all the answers

Evaluate ln b if ln a = 0.1 and ln b = 0.3, and find ln(a/b).

-0.2 Signup and view all the answers

Match the following options with their corresponding statements regarding polynomial characteristics:

A = Leading coefficient is positive and degree is 7 B = Leading coefficient is positive and degree is 4 C = Leading coefficient is negative and degree is 3 D = Leading coefficient is negative and degree is 6 E = Leading coefficient is negative and degree is 5 Signup and view all the answers

For the polynomial f(x) = x² + bx + c with zeros at x = 1 + √3 and x = 1 − √3, the value of b is ______.

-2 Signup and view all the answers

What is the value of sec θ if the adjacent side is 3 and the hypotenuse is √13?

√13/3 Signup and view all the answers

Match the functions with their properties:

f(x) = x^6 = Even function f(x) = 2 sin x + 6x = Odd function f(x) = cos x − 3x = Odd function f(x) = e^x − x^4 = Neither f(x) = 3x^2 + 7 = Even function Signup and view all the answers

Evaluate tan(5π/6).

−√3 Signup and view all the answers

Solve for x in the equation 2 log₇(x) + 5 = 9.

49 Signup and view all the answers

The function f(x) = 3x^2 + 7 is an odd function.

False Signup and view all the answers

For the polynomial f(x) = 3x^2 + 1, what is the simplified expression for (f(x + h) - f(x)) / h?

6x + 3h Signup and view all the answers

The expression ln(e) equals 1.

True Signup and view all the answers

Find the inverse of the function f(x) = (7 - 3x).

f^(-1)(x) = (7 - x)/3 Signup and view all the answers

Match the following trigonometric evaluations:

tan(5π/6) = −√3 sec(3π) = -1 sin(−225°) = −√2/2 Signup and view all the answers

What is the inverse function of $f(x) = x + 5$?

f −1(x) = x − 5 Signup and view all the answers

3 sec x = 3 sin x is a true identity.

False Signup and view all the answers

Solve for x: $2 cos x + cos x = 0$.

x = 0, π, 3π/2 Signup and view all the answers

The inverse function of $f(x) = 3x - 7$ is $f^{-1}(x) = ____$.

7 + 3x Signup and view all the answers

Which of the following values of x satisfies the equation $e^x - 3e - 10 = 0$?

x = ln 5 only Signup and view all the answers

The equation $y = 2 - e^x$ has a graph that decreases as x increases.

True Signup and view all the answers

The derivative of the function _____ is equal to $2 - e^x$.

y = 2 - e^x Signup and view all the answers

Match the following inverse functions with their corresponding forms:

$f^{-1}(x) = x - 5$ = $f(x) = x + 5$ $f^{-1}(x) = 7 + 3x$ = $f(x) = 3x - 7$ $f^{-1}(x) = 5x - 3$ = $f(x) = x - 3$ $f^{-1}(x) = 7 - 5x$ = $f(x) = 5x + 7$ Signup and view all the answers

Study Notes

Exam Instructions - Part 1

Calculators are not allowed.
Write name, instructor/section, and signature.
Show all work for full credit.
Circle the correct answer for multiple choice (Problems 1-18).
Mark the answers on the Scantron sheet.
Write answers to free response problems (Problems 19-24) in the provided space.
Time limit: 90 minutes.
Part 2 will follow and contains additional problems.
Answers on the Scantron cannot be changed once collected.

Exam Instructions - Part 2

Calculators are allowed.
Write name, instructor/section, and signature.
Show all work for full credit.
Circle the correct answer for multiple choice (Problems 101-106).
Mark the answers on the Scantron sheet (back side).
Write answers to free response problems (Problems 107-110) in the provided space.
Time limit: 30 minutes.

Problem 1 (Graphing & Domain & Range)

Identify domain (set of x-values) and range (set of y-values) of f(x).
Graph is provided.

Problem 2 (Parallel Lines)

Find the equation of a line parallel to a given line.
The line passes through a given point.

Problems 3 (Polynomial Degree & Leading Coefficient)

Determine degree and leading coefficient of polynomial.
Graphs of polynomials are given.

Problem 4 (Difference Quotient)

Determine and simplify the difference quotient for a function.

Problem 5 (Trigonometry in Right Triangle)

Evaluate trigonometric function (sec θ) given right triangle.

Problem 6 (Domain of Function)

Determine the domain of a given function.
Write the domain in interval notation.

Problem 7 (Trigonometry (Cosine Function))

Determine A and B in the equation f(x) = A cos(Bx).
Graph is given.

Problem 8 (Asymptotes of a Function)

Find the asymptotes of a given rational function.

Problem 9 (Logarithms)

Evaluate a logarithm given values of ln a and ln b.

Problem 10 (Odd Functions)

Identify odd functions from a list of functions.

Problem 11 (Inverse Functions)

Find the inverse function f⁻¹(x) for a given function f(x).

Problem 12 (Trigonometry)

Simplify a given trigonometric expression.
Consider the domain of the angle.

Problem 13 (Solving for x (exponents))

Solve for x in the exponential equation.

Problem 14 (Solving for x (trigonometry))

Find all x values that satify a given trigonometric equation in a specific interval.

Problem 15 (Graphing (exponential))

Identify the graph of a given function.

Problem 16 (Solving inequality)

Solve the inequality (function with powers)

Problem 17 (Trigonometry (Inverse Cosine))

Evaluate the inverse cosine function with a given value

Problem 18 (Rectangle perimeter)

Find a function to represent the perimeter of a rectangle.
Rectangle is formed with given information.

Problem 19 (Trigonometry (tan θ))

Determine tan θ given cos θ.

Problem 20 (Solving for x (fractional))

Solve for x using fractional expression.

Problem 21 (Solving for x (logarithm))

Solve for x involving a logarithmic equation.

Problem 22 (Polynomial (zeros))

Find the value of b in a polynomial given its zeros.

Problem 23 (Natural Logarithm)

Evaluate the expression that has ln(..)

Problem 24 (Trigonometry evaluation)

Evaluate trigonometric functions (tan(..), sec(..), sin(..)) for given arguments.

Problem 101 (Linear Equation)

Find a linear equation to describe price of a cup of coffee.

Problem 102 (Composite Functions)

Evaluate a composite function.

Problem 103 (Compound Interest)

Calculate initial investment to reach a future value using continuous compounding

Problem 104 (Angle of Elevation)

Determine the angle of elevation given two lengths (height, distance).

Problem 105 (Piecewise Function graph)

Match a piecewise function to its graph

Problem 106 (Quadratic equation, vertex)

Find the standard form of quadratic equation for parabola given vertex and point.

Problem 107 (Solving Exponential Equation)

Solve for x in an exponential equation. Round answer.

Problem 108 (Solving Trigonometric Equation)

Solve for x for a trigonometric equation within a specified domain. Round answer.

Problem 109 (Logarithm)

Evaluate a logarithm of given value.

Problem 110 (Projectile height)

Find the maximum height of a projectile given a function for projectile's height over time.

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Description

This document provides instructions for a mathematics exam divided into two parts. It details the use of calculators, answer marking procedures, and guidelines for showing work for maximum credit. Each part has specific problem types and time limits.

Mathematics Exam Instructions - Parts 1 & 2

Choose a study mode

Podcast

Questions and Answers

What is the price of a cup of coffee 10 years after the cafe opened?

The linear equation P = 0.04t + 2.25 correctly describes the price of coffee at the cafe.

What is the formula for the price of coffee in terms of the number of years since the cafe opened?

If Alice wants to have $2,000 after 5 years at an interest rate of 3% compounded continuously, which option is her initial investment?

The function g(x) = e^{2x} represents _______.

Calculate g ◦ f(169). What is the rounded result to the nearest hundredth?

The cafe opened with a cup of coffee priced at $3.05.

Match the financial terms with their definitions:

Which of the following inequalities has the solution set described by the interval notation (−∞, −2] ∪ [0, 1]?

The interval notation (−∞, −2) indicates that -2 is included in the solution set.

What is the general form of the solution set for the inequality x^3 + x^2 ≥ 2x?

The range of inputs for the arccos function is from ______.

Match the following expressions with the correct descriptions:

What is the correct output of the function arccos(√−3)?

The cubic polynomial x^3 + x^2 can be factored as x^2(x + 1).

What are the critical points of the inequality x^3 + x^2 - 2x ≥ 0?

What is the correct equation represented by option (A)?

Which of the following represents the union of intervals excluding 0 and -2?

The degree of a polynomial affects the number of roots it can have.

Evaluate the difference quotient for f(x) = 3x^2 + 1.

The function f(x) = A cos(Bx) can have A as a negative value.

What is the perimeter function P(x) for the rectangle formed by a point in the first quadrant on the line y = 6 − 2x?

In a right triangle, sec θ is the reciprocal of the ______.

What are the values of A and B if the function is represented as f(x) = 3 cos(2x)?

The function P(x) = 12x − 6 represents the perimeter of the rectangle formed by a point (x, y) on the line y = 6 − 2x.

If cos θ = 3/5 for θ in Quadrant IV, what is the value of tan θ?

Which of the following statements about the degree and leading coefficient of a polynomial is true?

The asymptotes for the function y = (3x − 3)/(x^2 − 8x + 16) are ______.

Evaluate ln b if ln a = 0.1 and ln b = 0.3, and find ln(a/b).

Match the following options with their corresponding statements regarding polynomial characteristics:

For the polynomial f(x) = x² + bx + c with zeros at x = 1 + √3 and x = 1 − √3, the value of b is ______.

What is the value of sec θ if the adjacent side is 3 and the hypotenuse is √13?

Match the functions with their properties:

Evaluate tan(5π/6).

Solve for x in the equation 2 log₇(x) + 5 = 9.

The function f(x) = 3x^2 + 7 is an odd function.

For the polynomial f(x) = 3x^2 + 1, what is the simplified expression for (f(x + h) - f(x)) / h?

The expression ln(e) equals 1.

Find the inverse of the function f(x) = (7 - 3x).

Match the following trigonometric evaluations:

What is the inverse function of $f(x) = x + 5$?

3 sec x = 3 sin x is a true identity.

Solve for x: $2 cos x + cos x = 0$.

The inverse function of $f(x) = 3x - 7$ is $f^{-1}(x) = ____$.

Which of the following values of x satisfies the equation $e^x - 3e - 10 = 0$?

The equation $y = 2 - e^x$ has a graph that decreases as x increases.

The derivative of the function _____ is equal to $2 - e^x$.

Match the following inverse functions with their corresponding forms:

Study Notes

Exam Instructions - Part 1

Exam Instructions - Part 2

Problem 1 (Graphing & Domain & Range)

Problem 2 (Parallel Lines)

Problems 3 (Polynomial Degree & Leading Coefficient)

Problem 4 (Difference Quotient)

Problem 5 (Trigonometry in Right Triangle)

Problem 6 (Domain of Function)

Problem 7 (Trigonometry (Cosine Function))

Problem 8 (Asymptotes of a Function)

Problem 9 (Logarithms)

Problem 10 (Odd Functions)

Problem 11 (Inverse Functions)

Problem 12 (Trigonometry)

Problem 13 (Solving for x (exponents))

Problem 14 (Solving for x (trigonometry))

Problem 15 (Graphing (exponential))

Problem 16 (Solving inequality)

Problem 17 (Trigonometry (Inverse Cosine))

Problem 18 (Rectangle perimeter)

Problem 19 (Trigonometry (tan θ))

Problem 20 (Solving for x (fractional))

Problem 21 (Solving for x (logarithm))

Problem 22 (Polynomial (zeros))

Problem 23 (Natural Logarithm)

Problem 24 (Trigonometry evaluation)

Problem 101 (Linear Equation)