If a verb in the following sentences is incorrect, cross it out and write the correct form above it. If the verb is already correct, write C above it.

Example 1. The chef $\overset{\textit{\color{#c34632}{set}}}{\sout{\text{sat}}}$ an assortment of ingredients on the counter.

She had set on a high stool at the work counter.

In the following exercise, sketch the graph of the plane curve given by the vector-valued function and, at the point on the curve determined by $r\left(t_0\right)$, sketch the vectors $T$ and $N$. Note that $\mathbf{N}$ points toward the concave side of the curve.

Function: $\mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}$, Time: $t_0=\pi$

For the following exercise, refer to the table and solve.

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8.7 & 12.3 & 15.4 & 18.5 & 20.7 & 22.5 & 23.3 & 24 & 24.6 & 24.8 \\ \hline \end{array}$$

To the nearest whole number, what is the predicted carrying capacity of the model?

A lens has one convex surface of radius $6.00 \mathrm{~cm}$ and one concave surface of radius $10.0 \mathrm{~cm}$. When an object is placed $35.0 \mathrm{~cm}$ from the lens, a real image is formed $50 \mathrm{~cm}$ from the lens.
(a) What is the focal length of the lens?
(b) What is the index of refraction of the lens?

Label each of the following functions, g(x), as a vertical stretch or a vertical compression of the parent function, f(x), and tell whether it is reflected about the x-axis. $g(x)=-1.2(5)^{x}, f(x)=5^{x}$

Use the table along with dimensional analysis, to convert the given unit to the unit indicated. Where necessary, round to two decimal places.

$$\begin{matrix} \text{Volume in Cubic Units} & \text{Capacity}\\ \text{1 }{\mathrm{cm}^{3}} & \text{ = 1 mL}\\ \text{1 }{\mathrm{dm}^{3}=1000 \mathrm{cm}^{3}} & \text{ = 1 L}\\ \text{1 }{\mathrm{m}^{3}} & \text{ = 1 kL}\\ \end{matrix}$$

2.6 L to c

Acetic acid, $\mathrm{HC}_2 \mathrm{H}_3 \mathrm{O}_2$, is responsible for the sour taste of vinegar. Combustion of acetic acid gives off $14.52 \mathrm{~kJ} / \mathrm{g}$ of heat. When $15.00 \mathrm{~g}$ of acetic acid is burned in a bomb calorimeter (heat capacity $=2.166 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}$ ) containing $0.800 \mathrm{~kg}$ of water, the final temperature in the calorimeter is $64.14^{\circ} \mathrm{C}$. What was the initial temperature of the calorimeter?

Some populations initially grow exponentially but eventually level off. Equations of the form $P(t)=\frac{M}{1+A e^{-k t}}$ where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. Here M is called the carrying capacity and represents the maximum population size that can be supported, and $A=\frac{M-P_{0}}{P_{0}}$, where $P_0$ is the initial population. Compute $\lim _{M \rightarrow \infty} P(t)$. What kind of function is your result?

Given the function $f(x)=2 x^3+3 x^2-12 x-7$, complete these steps:

a. Graph using $[-10,10] 1$ by $[-10,10] 1$ and $[-10,10] 1$ by $[-20,20] 2$.

b. Fill in this table:

c. Approximate the $x$ intercept and $y$ intercept to two decimal places.

d. Find the relative maximum and relative minimum points.

e. Find the intervals over which $f(x)$ is increasing.

f. Find the intervals over which $f(x)$ is decreasing.

g. Find any inflection points.

h. Find the intervals over which the graph of $f(x)$ is concave upward.

i. Find the intervals over which the graph of $f(x)$ is concave downward.

j. Show that the concavity changes from upward to downward, or vice versa, when $x$ moves from a little less than the point of inflection to a little greater than the point of inflection.

k. Find the largest and smallest values for this function for $-4 \leq x \leq 2$.

Let $I _ { m , n } = \int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } d x$ for constant m, n. Show that $I _ { m , n } = I _ { n , m }.$

Elliptical concrete pipes have a greater capacity for shallow flow than circular pipes. For this reason, elliptical pipes are often used in storm drains, culverts, and sewers. An engineer wants to design an elliptical concrete pipe with a maximum horizontal opening of 4 ft and a maximum vertical opening of 3 ft. a. Write an equation for an elliptical cross section of the pipe. For convenience, place the coordinate system with (0,0) at the center of the pipe. b. To construct the pipe, the engineer needs to know the location of the foci. How far from the center are the foci?

Some organisms exhibit a density-dependent mortality from one generation to the next. Let R > 1 be the net reproductive rate (that is, the number of surviving offspring per parent), let x > 0 be the density of parents, and y be the density of surviving offspring. The Beverton-Holt recruitment curve is $y=\frac{R x}{1+\left(\frac{R-1}{K}\right) x}$ where K > 0 is the carrying capacity of the organism’s environment. Show that $\frac{d y}{d x}>0$, and interpret this as a statement about the parents and the offspring.

At the instant shown, the 11,000-kg airplane's velocity is $\mathbf{v}=3001(\mathrm{~m} / \mathrm{s})$. The rate of change of the magnitude of the velocity is $d v / d t=5 \mathrm{~m} / \mathrm{s}^2$. The radius of curvature of the airplane's path is $4500 \mathrm{~m}$, and the $y$ axis points toward the concave side of the path. The thrust is $T=120,000 \mathrm{~N}$. Determine the lift $L$ and drag $D$.

While each state has two senators, members of the House are apportioned $\underline{\phantom{\text{justTTTtext}}}$.

a. according to the state's geographic size

b. based on the state's economic size

c. according to the state's population

d. based on each state's need