Math 131 Exam 1 Practice Exercises
Tom Linton, Central College, Spring 2000 The answers can be seen here.

1. Sketch a graph (using your calculator) of f(x) = | 3cos(2x - 1) + 2 | = abs(3cos (2x - 1) + 2). What is the period?
2. The graph of a function f(x) is shown below.

1. On which x-intervals is f(x) decreasing?
2. What is f(3)? For which x-value(s) does f(x) = 0?
3. On which x-intervals is f(x) concave up? Estimate the coordinates of the inflection point of f(x).
4. Where (which x-value) does f(x) have a valley bottom (local minimum).
3. Shown below is a plot of the temperature (in degrees Fahrenheit) of a warm beverage which was put in a cold refrigerator. The input is time measured in minutes, with t = 0 corresponding to the time when the beverage was put in the refrigerator. One (exact) point on the graph is t = 2, Temp = 75 and the formula for the beverage's temperature has the form B(t) = c + a* b^t (a, b, c are constants, t  is the variable). Use the plot to answer the following questions.

1. The graph of B(t) is the graph of an exponential function, f(t) = a*b^t that has been shifted up by c units. When the base, b, is between zero and one, exponential functions approach the x-axis (or t-axis here), i.e. their graphs approach the line y = 0. This means that B(t) will approach the line y = c (the exponential part gets close to zero). Use these facts to guess the value of c?
2. Since (0,80) and (2,75) are on the graph of B(t), the points (0,80-c) and (2,75-c) are on the graph of the exponential function f(t) = a*b^t. Show that f(2) / f(0) = b², (75-c) / (80-c) = f(2) / f(0) and that f(0) = a. Use these facts (and your value of c from part a) to estimate values for b and a.
3. You should now have numeric values for a, b, and c. Plot the function B(t) = c + a* b^t (on your calculator, using the window of the plot above) to make sure that your values of these constants are reasonable. Once your plot looks like the one above, add a graph of Y₂ = 50 and use the intersection feature of your calculator to find the time, t, with B(t) = 50.
4. Use algebra to find the time t with B(t) = 50. That is, solve c + a*b^t = 50 for t.
5. The last two questions asked for a time t with B(t) = 50. These questions are identical to asking for a single value of the inverse function B^-1. What value of B^-1 is asked for in the last two questions?
6. Find B^-1(40).
4. An algebraic representation of the function f(x) and a plot of the function g(x) are given below. Use these representations of the functions to answer the following questions.

 

f(x) =

4, 4 - 3x,

if x <= 0 if x > 0


1. What is f(1)? How about g(1)?
2. Find a value of x with f(x) = 2.
3. What value of x has g(x) = -20?
4. Let h(x) = g( f(x) ). What is h(1/3)?
5. What value of x has h(x) = -20?
5. A graph of 4x³ - 1.9^x  for 0 < x < 25, has two zeros and a hilltop (local maximum). Find accurate estimates of where these things occur.
6. The graph below has the form f(x) = A*sin (Bx + C) + D, for some values of the constants A, B, C, D. Find values for these constants.