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:

[Maple Math] = [Maple Math]

Hence, the answer is (C) .

2. We multiply numerator and denominator by [Maple Math] , take into account that [Maple Math] and obtain [Maple Math] . 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)

4. [Maple Math] (C)

5. By distributivity [Maple Math] . That gives (A).

6. We can factor out x and expand [Maple Math] . (B).

7. By expanding [Maple Math] and writing the denominator as [Maple Math] we obtain (B).

8. The lowest common denominator is [Maple Math] . Hence

[Maple Math]

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 [Maple Math] , 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. [Maple Math] 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 [Maple Math] .

12. (B)

13. [Maple Math]

14. [Maple Math] , so f(10)=-99=f(-10). [Maple Math] . Expanding the square and simplifying, we obtain [Maple Math] .

15. Increasing in ( [Maple Math] ) and ( [Maple Math] ), decreasing in (-1,1). Never constant.

16. Relative minimum at [Maple Math] , maximum at [Maple Math] .

17. The function is odd as clearly [Maple Math] , or, in other words, the graph is symmetric with respect to the origin.

18. [Maple Math] , 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 [Maple Math] . That is, [Maple Math] . The graphs of g(x) and h(x) look as follows:

[Maple Plot]

21. Solving for y gives us [Maple Math] . Hence, the slope is [Maple Math] , y-intercept 3.

22. The slope of a perpendicular line is the negative reciprocal of the slope of the original line. In our case, [Maple Math] . Since the line passes through (0,0), the y intercept is 0. Hence, the equation is [Maple Math] .

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.

25. y=0.15625x

26. 12

27. [Maple Math]

28. (a) 1+3i (b) 1+2i