Let f(x)=x^3-3 x^2-1, $x \geq 2$ Find the value of df^{-1} / dx at the point x=-1=f(3)

Graph each function. Label the axis of symmetry and the vertex.

$$y = x ^ { 2 } + 4 x + 3$$

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable. $x=y^{x y}$

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable: $y=(x+1)^{x}$

Let $f(x)=\frac{e^x}{1+e^{2 x}}$.

Find all absolute extreme values for f.

Using a suitable contour integral, evaluate the following real integrals:

$$\begin{array}{l}{\text { (a) } \int_{-\infty}^{\infty} \frac{\mathrm{d} x}{x^{2}+x+1} \quad \text { (b) } \int_{-\infty}^{\infty} \frac{\mathrm{d} x}{\left(x^{2}+1\right)^{2}}} \\ {\text { (c) } \int_{0}^{\infty} \frac{\mathrm{d} x}{\left(x^{2}+1\right)\left(x^{2}+4\right)^{2}}}\end{array}$$

(d) $\int_{0}^{2 \pi} \frac{\cos 3 \theta}{5-4 \cos \theta} \mathrm{d} \theta \quad(\mathrm{e}) \int_{0}^{2 \pi} \frac{4 \mathrm{d} \theta}{5+4 \sin \theta}$

$$\begin{array}{l}{\text { (f) } \int_{-\infty}^{\infty} \frac{x^{2} \mathrm{d} x}{\left(x^{2}+1\right)^{2}\left(x^{2}+2 x+2\right)}} \\ {\text { (g) } \int_{0}^{2 \pi} \frac{\mathrm{d} \theta}{3-2 \cos \theta+\sin \theta} \quad \text { (h) } \int_{0}^{\infty} \frac{\mathrm{d} x}{x^{4}+1}}\end{array}$$

$$\begin{array}{l}{\text { (i) } \int_{-\infty}^{\infty} \frac{\mathrm{d} x}{\left(x^{2}+4 x+5\right)^{2}}} \\ {\text { (j) } \int_{0}^{2 \pi} \frac{\cos \theta}{3+2 \cos \theta} \mathrm{d} \theta}\end{array}$$

Find twopossible functions $F$ given the third-order derivatives.

$$f^{\prime \prime \prime}(x)=8 e^{-2 x}-\sin x$$

A firm is profitable if A. TR < TC. B. AR < ATC. C. MC < ATC. D. ATC < P. E. ATC > MC.

Write an equation of the line that passes through each pair of points. (-2, -4), (2, 4)

Given f(x) = 2x^2, $x\geq 0$, a = 5: (a) Find f^{-1}(x). (b) Graph f and f^{-1} together. (c) Evaluate df/dx at x = a and df^{-1}/dx at x = f(a) to show that at these points df^{-1}/dx = 1/(df/dx).

Find the limit or show that it does not exist. lim(x→∞) = 1- x^2 / x^3-x+1

Find the area of each circle to the nearest tenth. Use 3.14 for $\pi.$ r = 7.2 yd

Which phrase best describes a group or family in the periodic table? (a) elements that have the same melting point (b) elements that have the same number of protons in their nucleus (c) elements that are all in the same physical state at room temperature (d) elements that have the same number of electrons in their outermost orbits

Use graphical differentiation to verify that $\frac{d}{d x}\left(e^{x}\right)=e^{x}$ .