If you have difficulty accessing these materials due to visual impairment, please email me at mohlenka@ohio.edu; an alternative format may be available.
The tests are cumulative and can include Pre-Calculus material.
For each section, we give
which exercises in that section make good test questions,
samples of other types of test questions, and
further information as needed.
Solutions
to the sample problems are not posted because posting them would
encourage you to settle for familiarity (the posted solution
makes sense) rather than mastery (you are sure your own solution
is correct).
I recommend you work in this order:
Read the book and understand the explanations and examples.
Do the text homework and online homework, which have answers available.
Look at this guide and do the additional sample problems.
This guide will be updated as we go along, so check it soon before each test.
Test 1, September 5
PreCalculus
Fractions and exponents
Determine whether each of the following statements is True or False.
\(\frac{A+B}{A+C}=\frac{B}{C}\)
\(\frac{A+B}{C+D}=\frac{A}{C}+\frac{B}{D}\)
\(x+x^{-1}=0\)
\((x+y)^2 = y^2+x^2+2xy\)
\(x^3 + y^3 = (x+y)(x^2 + xy + y^2)\)
Use the properties of exponents to simplify
\[ \left(\frac{25}{4x^4y^5}\right)\left(\frac{5}{2x^3y^2}\right)^{-3}\,.\]
Lines
Write the equation of the line passing through the two points \((1,3)\) and \((3,4)\).
Consider \(y=f(x)\) with graph:
Find the equations of the lines that comprise the graph of \(f\).
Draw the graph of \(g(x)=2f(-x)+1\). Mark and label the
points corresponding to \(A\), \(B\), and \(C\).
Trigonometry
Determine whether each of the following statements is True or False.
\(\frac{\sin(x^3)}{\sin(x^2)}=\frac{x^3}{x^2}=x\)
\(\sin(x+y)=\sin(x)+\sin(y)\)
Verify the identity
\(\frac{1}{1-\cos(\theta)}+\frac{1}{1+\cos(\theta)} = 2
\csc^2(\theta)\).
Exponentials and Logarithms
Solve the following equation for \(x\):
\(\log_3(x-4)+\log_3(x+4) = 2\).
Use the properties of logarithms to write \(f(x) = 2 \ln(x - 3) +
\log_e (y + 2) - \ln(z)\) as a single logarithm.
Function operations
Determine whether each of the following statements is True or False.
\(\sqrt{x^2+y^2}=\sqrt{x^2}+\sqrt{y^2}=x+y\)
\(f(x)=x^{-1}\;\Rightarrow\;f(x+h)=x^{-1}+h\)
With \(f(x)=\tan x\), \(g(x)=\frac{x}{x-1}\), and \(h(x)=\sqrt[3]{x}\), compute \((f\circ g\circ h)(x)\).
Simplification methods
Simplify and cancel so that you can plug in the given value
without dividing by 0. Then plug in the value.
For \(x=2\), \( \frac{x^2+x-6}{x-2}=\)
For \(x=4\), \( \frac{\sqrt{x}-2}{x-4}=\)
For \(h=0\), \( \frac{(x+h)^2 -x^2}{h}=\)
For \(h=0\), \( \frac{(x+h)^{-1} - x^{-1}}{h}=\)
For \(h=0\) and \(f(x)=2x^2\), \(\frac{f(x+h) -f(x)}{h}=\)
Given a graph of \(f\), sketch a graph of \(f'\) and \(f''\).
Given a formula for \(f\), use the limit definition to compute \(f'\) and \(f''\).
For each part below, sketch the graph of a function that
satisfies all the conditions or explain why it is impossible to
satisfy all the conditions. Conditions given in terms of \(x\)
(such as \(f(x) > 0\)) apply for all \(x\), whereas conditions
given in terms of \(a\) (such as \(f'(a) > 0\)) apply for a
single value \(x=a\), which you should label.
Consider the function \(f(x)=\dots\) on the interval \([\dots,\dots]\).
Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers \(c\)
that satisfy the conclusion of Rolle's Theorem.
State the Mean Value Theorem (MVT) using the template below.
If ...
then ...
Consider the function \(f(x)=\dots\) on the interval \([\dots,\dots]\).
Verify that the function satisfies each of the
hypotheses of the Mean Value Theorem on the given interval. Then find all
numbers \(c\) that satisfy the conclusion of the MVT.
Consider the function \(f(x)=\dots\) on the interval \([\dots,\dots]\).
Use the Intermediate Value theorem to show it has at least one root.
Use the Mean Value theorem to show it has at most one root.
If \(f\) is a differentiable function with \(f(0)=2\) and
\(f'(x) \le 4\), then what is the maximum value \(f(3)\) could
be? What is the minimum value \(f(3)\) could be?