Practice Exam 1 -- Answers and Some Solutions
1. Since a power of a power corresponds to the product of exponents, and product of powers to the sum of exponents we obtain:
Hence, the answer is (C) .
2. We multiply numerator and denominator by , take into account that and obtain . We can cancel one y, and obtain (B) .
3. The expression can't be simplified. The numerator and the denominator have no common factors! (D)
5. By distributivity . That gives (A).
6. We can factor out x and expand . (B).
7. By expanding and writing the denominator as we obtain (B).
8. The lowest common denominator is . Hence
We regroup terms in the numerator and obtain (A) .
9. A quotient is 0 if and only if the numerator is 0. Hence, the equation is equivalent to , which has two solutions x=4 and x=-4. At x=1 the expression is undefined. Thus x=1 is not a solution. The answer is (C) .
10. means by the geometric interpretation of the absolute value that the distance of x from 2 is less than 2. Hence, x must be between 0 and 4.
11. By simple algebra we obtain the solution .
14. , so f(10)=-99=f(-10). . Expanding the square and simplifying, we obtain .
15. Increasing in ( ) and ( ), decreasing in (-1,1). Never constant.
16. Relative minimum at , maximum at .
17. The function is odd as clearly , or, in other words, the graph is symmetric with respect to the origin.
18. , f(2) we estimate to be about 2. We estimate the points where f(x)=0 as x=-1.8,0,1.8.
19. No. It does not pass the vertical line test.
20. The function is . That is, . The graphs of g(x) and h(x) look as follows:
21. Solving for y gives us . Hence, the slope is , y-intercept 3.
22. The slope of a perpendicular line is the negative reciprocal of the slope of the original line. In our case, . Since the line passes through (0,0), the y intercept is 0. Hence, the equation is .
23. If you put all the parentheses correctly, you obtained the correct answer 32.0478....
24. The function is even as f(-x)=f(x). f(2)=f(-2)=12.
28. (a) 1+3i (b) 1+2i