In Exercise, graph the exponential function.

$$f(x)=-3^x+4$$

Tell whether the table of values represents a linear, an exponential, or a quadratic function. Then write an equation for the function using the form $y=m x+b, y=a b^{x}, \text { or } y=a x^{2}$.

$$\begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline y & 4 & 8 & 16 & 32 & 64 \\ \hline \end{array}$$

Numerical Approximation Use the Midpoint Rule and the Trapezoidal Rule with n=4 to approximate $\pi$ where

$$\pi=\int_0^1 \frac{4}{1+x^2} d x \text {. }$$

Then use a graphing utility to evaluate the definite integral. Compare all of your results.

a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $x y=1, \quad 1 \leq y \leq 2 ; \quad y$-axis

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

In the polynomial $4 x^2 y+x^3 y^2+3 x^2 y^3+7 y$, the term $x^3 y^2$ has degree 5 and no numerical coefficient.

For the following problem, use the values of $\sin 15^{\circ}$ and $\cos 15^{\circ}$ from the previous problems, with appropriate simplification, to show that the numerical expressions are exact values of $\tan 15^{\circ}$ and $\cot 15^{\circ}$.

$$\tan 15^{\circ}=2-\sqrt{3}$$

Which part of a microscope uses the real image formed by the objective lens in the column of the microscope and forms from it an enlarged, virtual image?

A) condenser lens

B) the numerical aperture

C) ocular lens

D) iris

E) focusing knob

Numerical Reasoning Consider a triangle of area 2 bounded by the graphs of y=x, y=0, and x=2.
Divide the interval $[0,2]$ into $n$ subintervals of equal width and show that the endpoints are

$$0<1\left(\frac{2}{n}\right)<\cdots<(n-1)\left(\frac{2}{n}\right)<n\left(\frac{2}{n}\right) .$$

In this Exercise. Find b such that

$$f(x)=-4 x^2+b x+3$$

has a maximum value of 50.

Use the table given below, which lists the population of the United States at each 10-year census from 1790 to 2000 (in millions).

$$\begin{array}{|c|c|} \hline \text { Year } & \text { Pop. } \\ \hline 1790 & 3.93 \\ 1800 & 5.31 \\ 1810 & 7.24 \\ 1820 & 9.64 \\ 1830 & 12.86 \\ 1840 & 17.06 \\ 1850 & 23.19 \\ 1860 & 31.44 \\ 1870 & 38.56 \\ 1880 & 50.19 \\ 1890 & 62.98 \\ 1900 & 76.21 \\ 1910 & 92.23 \\ 1920 & 106.02 \\ 1930 & 123.20 \\ 1940 & 132.16 \\ 1950 & 151.33 \\ 1960 & 179.32 \\ 1970 & 203.30 \\ 1980 & 226.54 \\ 1990 & 248.71 \\ 2000 & 281.42 \\ \hline \end{array}$$

a. Use the Continuous Change Model to obtain a formula for the population $P(t)$ of the United States t years after 1790 , based on an annual growth rate of 2.9655%.

b. Compute the predicted population of the United States for the year 2020 based on the formula for $P(t)$ obtained in Part a.

c. Calculate the population predicted by the formula $P(t)$ in Part a for the given years.

i. 1850

ii. 1900

iii. 1950

d. For which period is the formula obtained in Part a a reasonably accurate predictor of the actual population of the United States?

Use the table given below, which lists the population of the United States at each 10-year census from 1790 to 2000 (in millions).

$$\begin{array}{|c|c|} \hline \text { Year } & \text { Pop. } \\ \hline 1790 & 3.93 \\ 1800 & 5.31 \\ 1810 & 7.24 \\ 1820 & 9.64 \\ 1830 & 12.86 \\ 1840 & 17.06 \\ 1850 & 23.19 \\ 1860 & 31.44 \\ 1870 & 38.56 \\ 1880 & 50.19 \\ 1890 & 62.98 \\ 1900 & 76.21 \\ 1910 & 92.23 \\ 1920 & 106.02 \\ 1930 & 123.20 \\ 1940 & 132.16 \\ 1950 & 151.33 \\ 1960 & 179.32 \\ 1970 & 203.30 \\ 1980 & 226.54 \\ 1990 & 248.71 \\ 2000 & 281.42 \\ \hline \end{array}$$

a. Write an equation for the United States population $P_n n$ years after 1790 based on an annual growth rate of 2.9655%.

b. Compare the predictions of this discrete model with the data in the table. When is the predicted value off by the smallest percent of the actual population? When is it off by the largest percent?

What is the maximum of f(x)=16x-x$^2$ and where does it occur?

Correct the word in italics. If the word is correct, write C./ _____Is it *alright *to wear a striped necktie with a plaid shirt?.

For each item below, write a sentence in which you use the word or word pair correctly.

revelation / prosperity