Use a graphing calculator, if possible, or a computer algebra system such as Mathematica to first define the following functions and then simplify the expression $\frac{f(x+h)-f(x)}{h}$. See the Technology Note after the said example for guidance.

$$f(x)=\dfrac{1}{x-1}$$

While the Petersen family was waiting for their table, 9 people left the restaurant and 15 people entered. Find the sum of -9 and 15 to determine the change in the nu.m ber of people in the restaurant. Use algebra tiles to model the situation.

Use a computer algebra system to determine the antiderivative that passes through the given point. Use the system to graph the resulting antiderivative. $\int \frac{\sin \theta}{(\cos \theta)(1+\sin \theta)} d \theta,(0,1)$

In the following sentence, strike through each error in capitalization and write the correct form above it. If the sentence contains no error in capitalization, write C after it.

Patricia is going to sign up for Calculus if the Algebra II class is full.

Algebra Triangle A B C has side lengths of $A B=x^2-x+5, B C=5 x-4$, and $A C=3 x^2-2 x-1$. If $\triangle A B C$ is isosceles where $A B=B C$, find the lengths of all sides.

Use a computer algebra system to graph f and to find f$^{\prime}$ and f$^{\prime \prime}$. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f.

$$f(x)=x-\tan ^{-1}\left(x^{2}\right)$$

Determine if the differential equation is separable. If it is, find a general solution (perhaps implicitly defined) and any singular solutions arising from the algebra of the separation of variables. If it is not separable, do not attempt a solution. $y+y^{\prime}=e^{x}-\sin (y)$

Use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. $\int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x$

Paul states that $(a+b)^{2}$ is equivalent to $a^{2}+b^{2}$. Joyce thinks that something is missing. Help Joyce show Paul that the two expressions are not equivalent. Explain using at least two different approaches: diagrams, algebra, numbers, or words.

A symbol is an object, person, or idea that represents something else, usually an abstract concept. For example, on the literal level, a flag is a piece of cloth. However, a flag can also be the symbol of a nation. In this story, several objects are used as symbols. Write a brief description of what the following items symbolized in this story.

the contents of the dead man's pocket, as envisioned by Tom in a police report about his death

Use a computer algebra system to find the maximum value of $\left|f^{\prime \prime}(x)\right|$ on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section 4.3.)

$$f(x)=e^{-x^2 / 2}, \quad[0,1]$$

$$\begin{array}{|l r|}\hline \text{Credit Report:} & \$50\\ \text{Loan origination fee:} & 2.4\% \text{ of loan}\\ \text{Abstract of title:} & \$215\\ \text{Attorney fees:} & \$200\\ \text{Property tax:} & \$122.50 \\ \hline \end{array}$$

The McLaughlins were granted a $75,000 mortgage loan. At the time of the closing, they paid the closing cost shown. What is the total of the closing costs?

Algebra Two chords of a circle, $\overline{D E}$ and $\overline{F G}$, intersect at $C$ with segment lengths
$C D=3, C E=x+2, C F=6-x$, and $C G=5$. Write and solve an equation for $x$.

Use numerical evidence to conjecture the value of the limit if it exists. Check your answer with your Computer Algebra System (CAS). If you disagree, which one of you is correct? $\lim _{x \rightarrow 0^{+}} x^{\ln x}$