Evaluate the limits. $\lim _{x \rightarrow 0} \frac{10^{x}-e^{x}}{x}$

Suppose that the trapezoidal rule is used to estimate the value of $\int_{0}^{0.2} \int_{-0.1}^{0.1} e^{x^{2}+2 y} d y d x$. Determine the smallest value of n so that the resulting approximation $T_{n, n}$ is accurate to within $10^{-4}$ of the actual value of the integral.

a) Graph $y=\tan \theta$ for $-2 \pi \leq \theta \leq 0$ b) State the equations of the asymptotes for the graph in part a).

Find AB, if possible.

$$A=\left[\begin{array}{rr} {-1} & {3} \\ {4} & {-5} \\ {0} & {2} \end{array}\right], B=\left[\begin{array}{ll} {1} & {2} \\ {0} & {7} \end{array}\right]$$

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Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The set of points $\left\{(x, y, z) | x^{2}+y^{2}=1\right\}$ is a circle.

Determine the equation of the transformed image after the transformations described are applied to the given graph. a) The graph of $y=2 \log _{5} x-7$ is reflected in the x-axis and translated 6 units up. b) The graph of y = log (6(x - 3)) is stretched horizontally about the y-axis by a factor of 3 and translated 9 units left.

Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$. If the Mean Value Theorem cannot be applied, explain why not. $f(x)=x^{6}$, [-1, 1]

Write a proof for each limit using the $\varepsilon-\delta$ definition of a limit. $\lim _{x \rightarrow 3}(4 x)=12$

For each of the following situations, decide what sampling method you would use. Provide an explanation of why you selected a particular method of sampling. a. Alarge automotive companywants to upgrade the software on its notebook computers. A survey of 1,500 employees will request information concerning frequently used software applications such as spreadsheets,word processing, e-mail, Internet access, statistical data processing, and so on.Alist of employees with their job categories is available. b. A hospital is interested in what types of patients make use of their emergency room facilities. It is decided to sample 10% of all patients arriving at the emergency room for the next month and record their demographic information along with type of service required, the amount of time patient waits prior to examination, and the amount of time needed for the doctor to assess the patient’s problem.

(a) Explain why the Intermediate Value Theorem gives no information about the zeros of the function $f(x)=\ln \left(x^{2}+2\right)$ on the interval [-2, 2]. (b) Use technology to determine whether f has a zero on the interval [-2,2].

Let f be a function for which $0 \leq f(x) \leq 1$ for all x in [0, 1]. If f is continuous on [0, 1], show that there exists at least one number c in [0, 1] such that f(c)=c.

Find the first- and second-order Taylor polynomials for the given function f at the given point a. $f(x, y, z)=1 /\left(x^{2}+y^{2}+z^{2}+1\right), \mathbf{a}=(0,0,0)$

A fungal spore a. contains an embryonic organism.b. germinates directly into an organism.c. is always windblown, because none are motile.d. is most often diploid.e. Both b and c are correct.

What factors does Paul Krugman identify that supported the expansion of international trade in the 1800s?