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. $\int_{0}^{\pi / 2} \cos ^{6} x d x$

Underline each simple subject once and each simple predicate twice. Label each preposition $prep$., each direct object $d.o$., and each indirect object $i.o$.
Ms. Watkins asked Jenny an algebra question.

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. $y+y^{\prime}=e^{x}-\sin (y)$

In Exercise 3e, you graphed an inequality involving lines with the following equations.

$$2x+3y=12$$

$$2x+y=8$$

Use algebra to find the intersection of the two lines. Then check your result by testing the values of $x$ and $y$ in each equation.

Write a double integral that represents the surface area of $z=f(x, y)$ over the region $R$. Use a computer algebra system to evaluate the double integral.

$f(x, y)=4-x^2-y^2$ $R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\}$

In Exercises I through $22$, find the critical points of the given fianctions and classify each as a relative maximum, a relative minimum, or a saddle point. (Note: The algebra in Exercises $19$ through $22$ is challenging.)

$$f(x, y)=x^3-4 x y+y^3$$

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? $\lim _{x \rightarrow 0^{+}} x^{\ln x}$

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

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.

Prove that the field k(x) of rational functions over k in the variable x is not a finitely generated k-algebra. (Recall that k(x) is the field of fractions of the polynomial ring k[x]. Note that k(x) is a finitely generated field extension over k.)

A Boolean algebra may also be defined as a partially ordered set with certain additional properties. Let $(B, \leqslant)$ be a partially ordered set. For any $x, y \in B,$ we define the least upper bound of x and y as an element z such that $x \leqslant z, y \leqslant z,$ and if there is any element $Z^{*}$

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.

$$\frac{dy}{dx}=\frac{2y}{\sqrt{16-x^2}},\ y(0)=2$$

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result.
$h(x)=\dfrac{1}{9}(3 x+1)^3, \quad\left(1, \dfrac{64}{9}\right)$

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.

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