Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(t)=3t+5, [1, 2]

Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $g(x)=x^{2}+e^{x}$, [0, 1]

Use the graph of f in the figure to evaluate the function or analyze the limit.

$$\lim _{x \rightarrow 1} f(x)$$

Use the graph of f in the figure to evaluate the function or analyze the limit.

d. $\lim _{x \rightarrow-1} f(x)$

Use the graph of f in the figure to evaluate the function or analyze the limit.

$$\lim _{x \rightarrow 3^{-}} f(x)$$

Use the graph of f in the figure to evaluate the function or analyze the limit.

$$\lim _{x \rightarrow 3} f(x)$$

Use the graph of f in the figure to evaluate the function or analyze the limit.

$\lim _{x \rightarrow-1^{+}} f(x)$

Use the graph of f in the figure to evaluate the function or analyze the limit.

$\lim _{x \rightarrow-1^{-}} f(x)$

Evaluate the triple integrals over the indicated region. Be alert for simplifications and auspicious orders of iteration. $\iiint_{D}(3+2 x y) d V$, over the solid hemispherical dome D given by $x^{2}+y^{2}+z^{2} \leq 4$ and $z \geq 0$

Solve the equation for $x$:

$$4 \sin x + 2 = 0; ~~~~~~ [0, 2π)$$

When the annual rate of inflation averages 5% over the next 10 years, the approximate cost C of goods or services during any year in that decade is $C(t)=P(1.05)^{t}$, where t is the time in years and P is the present cost. (a) The price of an oil change for your car is presently $29.95. Estimate the price 10 years from now. (b) Find the rates of change of C with respect to t when t=1 and t=8. (c) Verify that the rate of change of C is proportional to C. What is the constant of proportionality?

Find the flux of $\mathbf{F}=y^{3} z \mathbf{i}-x y \mathbf{j}+(x+y+z) \mathbf{k}$ across the portion of the surface $z = ye^x$ lying over the unit square $[0,1] \times[0,1]$ in the xy-plane, oriented by upward normal.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If (−4, −5) is a point on a graph that is symmetric with respect to the y-axis, then (4, −5) is also a point on the graph.

Determine whether $f$ is continuous at $c$.

$$f(x)=\left\{\begin{array}{cc}{4-3 x^{2}} & {\mathrm\ { if }\ x<0} \\{4} & {\mathrm\ { if }\ x=0} \\{\sqrt{16-x^{2}}} & {\mathrm\ { if }\ 0<x \leq 4}\end{array}\right.$$

at $c=0$.