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.

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

In each of the following problems, a. Write the form for the particular solution $y_{p}(x)$ for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $y^{\prime \prime}-y^{\prime}-2 y=x^{2} e^{x} \sin x$

Solve the problem. Your answer will be a ratio. Write the ratio as a fraction in lowest terms. An algebra test of 50 questions included 15 questions on Chapter 4. What is the ratio of questions from Chapter 4 to the other questions on the test?

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.

$$f(x)=\left(x^2+1\right)^3$$

Use a computer algebra system to find the derivative of the function. Use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \

$$f(x)=[(x-2)(x+4)]^2$$

Suppose you are using algebra tiles to represent each expression below. How many unit tiles would you need to add to build a perfect square? a. $x^{2}+10 x$ b. $x^{2}-6 x$ c. $x^{2}+22 x$

Use a computer algebra system to evaluate the following integral. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number. $\int_{0}^{\pi / 2} \cos ^{6} x d x$

The utility function U=f(x, y) is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is

$$U=-5 x^2+x y-3 y^2 .$$

Use a computer algebra system to graph the function. Interpret the marginal utilities of products $x$ and $y$ graphically.

For the half-Vee channel, let the interior angle of the Vee be $\theta$. For a provided value of the area, slope, and $n$, determine the value of $\theta$ for which the flow rate is a maximum. To avoid cumbersome algebra, simply plot $Q$ versus $\theta$ for constant $A$.

Slope Fields Use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition.

$$\frac{dy}{dx}=\frac{10}{x \sqrt{x^2-1}},\ y(3)=0$$

In given problem, find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit.

$$\lim _{x \rightarrow 2} \frac{x^2-4}{x^2+4}$$

Use algebra tiles to rewrite each relation in the form $y = a ( x - h ) ^ { 2 } + k$ by completing the square.

$$\begin{array} { l } a) { y = x ^ { 2 } + 2 x + 5 } \\ b) { y = x ^ { 2 } + 4 x + 7 } \\ c) { y = x ^ { 2 } + 6 x + 3 } \end{array}$$

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 $x$.

Algebra Two lines are perpendicular. One line has an equation of $r x+3 y=4$ and the other line has an equation of $\frac{2}{5} x+y=6$. Find a linear equation to describe the relationship between $r$ and $s$.