Exam 1: Functions and Limits

A box with an open top is to be constructed from a rectangular piece of card board with dimensions $b=4$ in. by $a=28$ in. by cutting out equal squares of side $X$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $X$ .

Simplify the expression. $\sin \left(3 \cos ^{-1} 5 x\right)$

Find the domain and sketch the graph of the function. What is its range? f (x) = $\left\{\begin{array}{ll}-x+2 & \text { if } x \geq 2 \\x^{2}-1 & \text { if } x<2\end{array}\right.$

Determine whether the function is even, odd, or neither. $f(x)=5 x^{3}+7 x$

Find the domain of the function. $f(x)=\frac{7 x+4}{x^{3}}$

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A measured in $m^{2}$ of the caves by the equation $S=0.7 A^{03}$ (a) The cave called mission impossible near puebla, mexico, has suface area of $A=90 \mathrm{~m}^{2}$ . How many species of bats would expect to find in that cave? (b) If you discover that 5 species of bats live in cave estimate the area of the cave.

The graph of the function $f$ follows. Choose the graph of $y=f(|x|) .$

Solve the equation. $2 e^{x+2}=5$

Use the laws of logarithms to write the expression as the logarithm of a single quantity. 4 ln 5 - $\frac{5}{6}$ ln (x + 6)

The graph of the function $f$ is given. State the value of $f(-0.4)$ .

Find all solutions of the equation correct to two decimal places. $\sqrt{x}=x^{3}-2$

Determine whether the function is one-to-one. $f(x)=\sqrt{1-x^{2}}$

Use the table to evaluate the expression $(f \circ g)(6)$ . $X$ 1 2 3 4 5 6 x 3 2 1 0 1 2 f(x) 6 5 2 3 4 6 g(x)

Determine whether f is even, odd, or neither. $f(x)=\frac{8 x^{2}}{x^{4}+5}$

Find the inverse of f. Then use a graphing utility to plot the graphs of f and $f^{-1}$ on the same set of axes. $f(x)=\frac{1-e^{8 x}}{1+e^{8 x}}$

Starting with the graph of $y=e^{x}$ , find the equation of the graph that results from reflecting about the line $y=3$ .

The following figure shows a portion of the graph of a function f defined on the interval $[-1,1]$ . Sketch the complete graph of f if it is known f is odd.

Refer to the graph of the function f in the following figure. a. Find f (1). b. Find the value of x for which (i) $f(x)=1$ and (ii) $f(x)=0$ . c. Find the domain and range of f.

Determine whether f is one-to-one.

The table gives the population of the United States, in millions, for the years 1900 - 2000. Use a graphing calculator with exponential regression capability to model the U.S. population since 1900. Use the model to estimate the population in 1965 and to predict the population in the year 2025. Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Population (millions) 76 92 106 123 131 150 179 203 227 250 281