Piecewise functions can sometimes be differentiable at a point where two pieces meet. For the function $f$, use one-sided limits to find the values of $a$ and $b$ that make $f$ differentiable at $x=2$.

$$f(x)=\left\{\begin{array}{l} a x^2+1, \quad \text { if } x<2 \\ -x^2+6 x+b, \quad \text { if } x \geq 2 \end{array}\right.$$

Plot the graph using Boolean variables to restrict the domain, and sketch the result.

Which aspects of capitalism do you personally appreciate? Which do you find worrisome?

$$\begin{array} { l } { \text { Determine the symmetry group of the tessellation of the plane ex- } } \\ { \text { emplified by the brickwork shown. } } \end{array}$$

Nettles, King, and Tanaka are partners sharing income 3:2:1. After the firm’s loss from liquidation is distributed, the capital account balances were: Nettles, $15,000 Dr.; King,$46,000 Cr.; and Tanaka, $71,000 Cr. If Nettles is personally bankrupt and unable to pay any of the$15,000, what will be the amount of cash received by King and Tanaka upon liquidation?

As a biologist, you wish to notice if there is a relationship between the heights of tall trees and their diameters. You see the following data for the diameter (in inches) of the tree at $4.5$ feet from the ground and the corresponding heights (in feet).

2. What type of nonparametric analysis could be used to answer the question?

A bag contains $2$ red, $6$ blue, $7$ yellow, and $3$ orange marbles. Once a marble is selected, it is not replaced. Find each probability.

$P(2$ yellows in a row then orange $)$

Complete the following table to compare polynomial approximations of values of cosine x to the actual values of cosine x.

$$\begin{matrix} \text{x} & \ {1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}} & \ {\cos x}\\ \text{0.1} & \text{ } & \text{ }\\ \text{0.2} & \text{ } & \text{ }\\ \text{0.3} & \text{ } & \text{ }\\ \end{matrix}$$

Answer each question about the surface area S on a surface given by a positive function z = f(x, y) over a region R in the xy-plane. Explain each answer. (a) Is it possible for S to equal the area of R?

A train's horn produces a $148-\mathrm{Hz}$ sound while moving at $32.0 \mathrm{~m} / \mathrm{s}$. What frequencies are heard by a person standing still beside the tracks as the train approaches and after it passes?

In given problem, $R=\{(x, y): 0 \leq x \leq 6,0 \leq y \leq 4\}$ and P is the partition of R into six equal squares by the lines x=2, x=4, and y=2. Approximate $\iint_R f(x, y) d A$ by calculating the corresponding Riemann sum $\sum_{k=1}^6 f\left(\bar{x}_k, \bar{y}_k\right) \Delta A_k$, assuming that $\left(\bar{x}_k, \bar{y}_k\right)$ are the centers of the six squares.

$$f(x, y)=\frac{1}{6}(48-4 x-3 y)$$

Describe two commitment problems you personally encountered during the last year.

A group of junior high students will be touring Washington, D.C. Their chaperons will have the $1,810 cost of the tour reduced by$15.50 for each student they personally supervise. How many students will a chaperon have to supervise so that his or her cost to take the tour will be $1,500?

In the independent and subordinate clauses in the given sentences, underline the subjects once and the verbs twice.

In 1909, funding was approved for this statue, which is located between Plymouth, Indiana, and the Yellow River.

Use the tabulated values off to estimate the value of $\int_{0}^{6} f(x) d x$ \int_{0}^{6} f(x) d x regular partition with n=3 subintervals.

$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline f(x) & {20} & {22} & {26} & {30} & {34} & {38} & {40} \\ \hline\end{array}$$