Exam 1: Functions and Models

Suppose pH = -log [H $+$ ]. Suppose further that for vinegar, the hydrogen ion concentration in moles per liter is given by [H $+$ ] $= 5.2 \left( 10 ^ { - 4 } \right)$ . Find the pH of the vinegar.

For the exponential function f (x) = c . a $x$ + b whose graph is given below, determine the values of a, b, and c.

Sketch a possible parametric curve defined by the following table of values. t 0 \pi 2\pi x 0 0.18 0 -0.53 -1 -0.88 0 1.24 2 y 0 0.18 0.5 0.53 0 -0.88 -1.5 -1.24 0

Classify the function f(x) = $\frac { \pi ^ { 2 } + x ^ { 2 } } { e }$

Relative to the graph of y = x $3$ , the graph of y = $\frac { 1 } { 2 }$ x $3$ is changed in what way?

Relative to the graph of y = x $2$ , the graph of y = x $2$ - 2 is changed in what way?

Determine an appropriate viewing rectangle for the graph of the function f (x) = 6x $2$ + x - 10.

Find the value of ln $\sqrt { e ^ { 3 } }$ .

Relative to the graph of y = sin x, the graph of y = 3 sin x is changed in what way?

Describe the curve defined by x = cos t, y = sin $2$ t.

The following table contains United States population data for the years 1790-2000 at intervals of 10 years. Year Years since 1790 U.S. population (millions) 1790 0 3.9 1800 10 5.2 1810 20 7.2 1820 30 9.6 1830 40 12.9 1840 50 17.1 1850 60 23.2 1860 70 31.4 1870 80 39.8 1880 90 50.2 1890 100 62.9 Year Years since 1790 U.S. population (millions) 1900 110 76.0 1910 120 92.0 1920 130 105.7 1930 140 122.8 1940 150 131.7 1950 160 150.7 1960 170 178.5 1970 180 202.5 1980 190 225.5 1990 200 248.7 2000 210 281.4 (a) Make a scatter plot for the data and use your scatter plot to determine a mathematical model for the U.S. population.(b) Use your model to predict the U.S. population in 2005.

Determine whether or not the function f (x) = $\frac { 3 x } { x - 1 }$ is one-to-one.

Determine an appropriate viewing rectangle for the graph of the function f (x) = 20 sin (40x) $+$ 10.

Determine an appropriate viewing rectangle for the graph of the function f (x) = 4x - $\left| 3 x ^ { 2 } - 10 \right|$ .

A function has a domain [-4, 4] and a portion of its graph is shown. (a) Complete the graph of f of its is known that f is an even function.(b) Complete the graph of f if it is known that f is an odd function.

Suppose that the position of one particle at time t is given by x $1$ = 4 sin t, y $1$ = 2 cos t for 0 $\leq t \leq 2 \pi$ and the position of a second particle is given by x $2$ = 2 cos t - 1, y $2$ = 4 sin t + 2 for 0 $\leq t \leq 2 \pi$ (a) How many points of intersection are there for these two paths? (b) Are any of these collision points (that is, points where the particles are at the same place at the same time)? If so, find the collision points.

Solve for $x : e ^ { 2 x - 1 } = e ^ { \operatorname { In } 6 }$ .

Suppose that P(t) represents the pressure (in pounds per square inch) in a tire with a slow air leak t minutes after the leak begins. In practical terms, what is: (a) P(20) (b) P $- 1$ (20)

Find $f ^ { - 1 } ( x )$ if $f ( x ) = \sqrt { e ^ { 2 } + 2 }$ .

Let f(x) = 2 - x $3$ and g(x) = 3 $+$ x. Find the value of $( f \circ g ) ( x )$ when x = - 5.