Math 12 (Spring 2001) Kouba Exam 1

Please PRINT your name here :

Your HW/Exam ID Number

1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO.

2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR CO-OPERATION.

3. YOU MAY NOT USE A CALCULATOR ON THIS EXAM.

4. No notes, books, or classmates may be used as resources for this exam.

5. Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

6. Put units on answers where units are appropriate.

7. You have until 9:50 a.m. sharp to finish the exam.

8. Make sure that you have 6 pages including the cover page.

1.) (8 pts.) Compute the distance between the points (2, −3) and (6, 0). 2.) (8 pts.) Find an equation of the line which is perpendicular to the line x + 2y = 5 and which passes through the point (1, 3). 3.) (9 pts. each) Solve each inequality for x.

a.) | 2x − 3 |< 7 b.) | x + 2 |≥ 5 4.) Consider the function f(x) = \sqrt{x − 4}.

a.) (5 pts.) Determine the domain of f. b.) (5 pts.) Determine the range of f.

5.) Let f(x) = \frac{x + 1}{x − 3} .

a.) (6 pts.) Show that f is one-to-one, i.e., show that if f(a) = f(b), then a = b.

b.) (8 pts.) Derive the inverse function y = f^{−1}(x).

6.) (10 pts.) Let f(x) = x^{2} −x+3. Simplify the expression \frac{f(x + h) − f(x)}{h}.

7.) (8 pts.) Use a sign chart to solve the following inequality for x : x^{3} − 25x ≤ 0

8.) (8 pts.) Let f(x) = x^{2} + 3 . Find a function y = g(x) so that (f ∘ g)(x) = 3x − 4.

9.) (8 pts.) Let f(x) = 5 + \sqrt{x}. Find the average rate of change of f on the interval [4, 9].

10.) (8 pts.) The graph of y = g(x) is given. Use translations and/or reflections to sketch the graph of y = −g(−x) + 2.

The following EXTRA CREDIT PROBLEM is worth 10 points. This problem is OPTIONAL.

1.) Assume that

1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + ^{...} + n^{3} = \frac{n^{2}(n + 1)^{2}}{4}

for any positive integer n. For example,

1^{3} + 2^{3} + 3^{3} + 4^{3} = \frac{4^{2}(4 + 1)^{2}}{4} = 100.

For what values of n is 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + ^{...} + n^{3} ≥ 6084 ? (HINT: \sqrt{24, 336} = 156. )