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.

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

Underline the correct pronoun. Identify the case by writing nom. (nominative) or obj. (objective) in the blank. Some sentences have more than one pronoun to identify./ _____These algebra problems confuse Rashonda as much as (I, me)

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

Algebra In $\triangle P Q R, \angle S Q R \cong \angle S R Q, P Q>P R, \mathrm{~m} \angle P S R=(4 y+9)^{\circ}$ and $\mathrm{m} \angle Q S P=(6 y-24)^{\circ}$. Find the range of values for $y$.

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.)

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

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

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

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. $x^{2}+22 x$

The $2003 ~Statistical ~Abstract ~of ~the ~United ~States$ reported the percentage of people $18$ years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of $.30.$

Assume that the study uses your sample size recommendation in part (a) and finds $520$ smokers. What is the point estimate of the proportion of smokers in the population?

Use a computer algebra system to determine the antiderivative that passes through the given point. Use the system to graph the resulting antiderivative. $\int \frac{x(2 x-9)}{x^{3}-6 x^{2}+12 x-8} d x \ (3, 2)$

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.)

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.)

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

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.

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 $Z^{*}$