Read the sentence below, and decide whether it is clear or it contains a dangling modifier. On the line provided, write C for clear or D for dangling modifier. Example $\underline{\text{D}}$ 1. Finally finding the old book, Gordon’s quest was over.

______ To solve the problem, patience is necessary.

Choose the best answer to given of the following. Explain your reasoning with one or more complete sentences.
We determine the distance of a Cepheid in another galaxy by
(a) measuring its parallax.
(b) determining its luminosity from the period-luminosity relation and then applying the inverse square law for light.
(c) knowing that all Cepheids have about the same luminosity and then applying the inverse square law for light.

Use Cauchy`s residue theorem, where appropriate, to evaluate the given integral along the indicated contours. $\oint_C \frac{z+1}{z^2(z-2 i)} d z$ (a) $|z|=1$ (b) $|z-2 i|=1$ (c ) $|z-2 i|=4$

Describe contours C for which we are guaranteed that $\oint_C f(z) d z=0$ for each of the following functions. (a) $f(z)=\frac{1}{z^3+z}$ (b) $f(z)=\csc z$ (c) $f(z)=\frac{1}{1-e^z}$ (d) $f(z)=\operatorname{Ln} z$

Think About It Consider a function $f$ such that $f$ is increasing. Sketch graphs of $f$ for (a) $f^{\prime}<0$ and (b) $f^{\prime}>0$.

Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f’ and f’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$$f(x)=4x^4-32x^3+89x^2-95x+29$$

Sketch the indicated curves and surfaces.

The surface of a small hill can be roughly approximated by the equation $z\left(2 x^2+y^2+100\right)=1500$, where the units are meters. Draw the surface of the hill and the contours for $z=3 \mathrm{~m}$, $z=6 \mathrm{~m}, z=9 \mathrm{~m}, z=12 \mathrm{~m}$, and $z=15 \mathrm{~m}$.

Let $f(x)=x^2$ for $0 \leq x \leq 1$. The average value of $f$ on $[0,1]$ is $\frac{1}{3}$. Find a nonnegative weight function $w$ such that the weighted average of $f$ on $[0,1]$, as defined by Equation (2.19) is

(a) $\frac{1}{2}$;

(b) $\frac{3}{5}$;

(c) $\frac{2}{3}$.

Evaluate each determinant.

$$\left| \begin{array} { l l l } { a } & { b } & { c } \\ { c } & { a } & { b } \\ { b } & { c } & { a } \end{array} \right|$$

Use a CAS to plot graphs of partial sums $\left\{S_{N}(x)\right\}$ of the given trigonometric series. Experiment with different values of N and graphs on different intervals of the x-axis. Use your graphs to conjecture a closed-form expression for a function f defined for $0<x<L$ that is represented by the series. $f(x)=-\frac{\pi}{4}+\sum_{n=1}^{\infty}\left[\frac{(-1)^{n}-1}{n^{2} \pi} \cos n x+\frac{1-2(-1)^{n}}{n} \sin n x\right]$

Clear fractions or decimals, solve, and check. \

$$0.76+0.21 t=0.96 t-0.49$$

If prism A and prism B have a ratio of similarity of 2:3, what is the volume of prism B if the volume of prism A is 36 cubic units?

Find equation s of two straight lines tangent to the graph of $y=\sin ^{-1} x$ and having slope 2.

Simplify the expression. Assume that all variables are unrestricted and use absolute value symbols when necessary

$$\sqrt [ 3 ] { 64 x ^ { 6 } y ^ { 3 } }$$