A. Fill in the Blanks_____________________________________________

In Problems 1–20, fill in the blanks.

1. Completing the square in x for 2x² + 6x + 5 gives ______.

2. In the binomial expansion of (1 – 2x)³ the coefficient of x² is ______.

3. In interval notation, the solution set of the inequality is ______.

4. If a – 3 is a negative number, then |a – 3| = ______.

5. If |5x| = 80, then x = ______.

6. If (a, b) is a point in the third quadrant, then (a, b) is a point in the ______ quadrant.

7. The point (1, 7) is on a graph in the Cartesian plane. Give the coordinates of another point on the graph if the graph is:

(a) symmetric with respect to the x-axis. ______

(b) symmetric with respect to the y-axis. ______

(c) symmetric with respect to the origin. ______

8. The lines 6x + 2y = 1 and kx – 9y = 5 are parallel if k = ______. The lines are perpendicular if k = ______.

9. The complete factorization of the function f(x) = x³ – 2x² – 6x is ______.

10. The only potential rational zeros of f(x) = x³ + 4x + 2 are ______.

11. The phase shift of the graph of y = 5 sin(4x + π) is ______.

12. If f(x) = x⁴ arctan (x/2), then the exact value of f(−2) is ______.

13. If sin x the sin 2x = ______.

14. If = 81, then x = ______.

15. arccos = ______.

16. 5 In 2 − In = In ______

17. The graph of y = In (2x + 5) has the vertical asymptote x = ______.

18. The domain of the function y = In (x² − 2x) is ______.

19. The number of five element subsets that can be formed from the set of letters in the English alphabet is ______.

20. If the first three terms of an arithmetic sequence are and a₁ = 10, a₂ = 6.5, a₃ = 3, then a₁₁ = ______.

B. True/False ____________________

In Problems 1–20, answer true of false.

  1. The absolute value of any real number is positive. ______

  2. The inequality |x| > −1 has no solutions. ______

  3. For any function f, if f(a) = f(b), then a = b. ______

  4. The graph of y = f(x + c) is the graph of y = f(x) shifted units to the right. ______

  5. The points (1, 3), (3, 11), and (5, 19) are collinear. ______

  6. The function f(x) = x⁵ = 4x³ + 2 is an odd function. ______

  7. is a factor of the function f(x) = 64x⁴ = 64x³ = 48x² − 36x − 12.______

  8. If b² − 4ac = 0, the graph of f(x) = ax² + bx + c, a ≠ 0, does not cross the x-axis. ______

  9. is a rational function. ______

10. If f(x) = x⁵ + 3x − 1, then there exists a number in [−1, 1] such that f(c) = 0. ______

11. The graph of the function f(x) has no x-intercepts. ______

12. x = 0 is a vertical asymptote for the graph of the rational function ______

13. The graph of y = cos(x/6) is the graph of stretched horizontally. ______

14. f(x) = csc x is not defined at x = π/2. ______

15. The function f(x) = e^−4x2 is not one-to-one.______

16. The exponential function f(x) = increases on the interval ______ (−∞, ∞)

17. The domain of the function f(x) = In x + In (x − 4) is (4, ∞) ______

18. The solutions of the equation In x² = In3x are x = 0 and x = 3. ______

19. cos²x + cos²(x – π/2) = 1 ______

20. In |cscx| + In |sin x| = 0______

C. Exercises ___________________________________________________

  1. Match the given interval with the appropriate inequality.

(i)   [2, 4]

(ii)  [2, 4]

(iii) (2, 4)

(iv) (2, 4]

(a) |x − 3 | ≤ 1

(b) 1 < x − 1 ≤ 3

(c) − 2 < 2 − x ≤ 0

(d) |x − 3| < 1

  2. Write the solution of the absolute-value inequality |3x − 1| > 7 using interval notation.

  3. The answer to a problem given in the back of a mathematics text is 1 + but your answer is 2/( − 1) Are you correct?

  4. In which quadrants in the Cartesian plane is the quotient x/y negative?

  5. Which one of the following equations best describes a circle that passes through the origin? The symbols a, b, c, d, and e stand for different nonzero real constants.

(a) ax² + by² + cx + dy + e = 0

(b) ax² + ay² + cx + dy + e = 0

(d) ax² + ay² + cx + dy = 0

(e) ax² + ay² + e = 0

(f) ax² + ay² + cx + e = 0

  6. Match the given rational function f with the most appropriate phrase.

(a) slant asymptote

(b) no asymptotes

(c) horizontal asymptote

(d) vertical asymptote

  7. What is the range of the rational function f(x) =

  8. What is the domain of the function f(x) = ?

  9. Find an equation of the line that passes through the origin and through the point of intersection of the graphs of x + y = 1 and 2x − y = 7.

10. Find a quadratic function whose graph has the y-intercept (0, −6) and the vertex of the graph is (1, 4).

In calculus you are often required to rewrite a function either in a simpler form or in a form that is more helpful in solving the problem. In Problems 11–16, rewrite each function by following the given instruction. In calculus you would be expected to recognize what to do from the context of the actual problem.

11. Express f as a quotient using rationalization and simplification.

12. Carry out the indicated division and express each term as a power of x.

13. Decompose f into partial fractions.

14. Express f in terms of sec x and tan x.

15. f(x) = e^3Inx Express as a power of x

16. Express f without absolute value signs.

In calculus you are often required to find zeros of a function. In Problems 17 and 18, solve the equation by following the given instruction.

17. . Rewrite f as a single expression without negative exponents.

18. f(x) = 2 sin x cos x − sin x. Find the zeros of on the interval [−π, π].

In Problems 19 and 20, compute and simplify the difference quotient for the given function.

21. Consider the trigonometric function y = −8 sin(πx/3). What is the amplitude of the function? Give an interval over which one cycle of the graph is completed.

22. If tanθ = and π < θ < 3π/2, then what is the value of cosθ?

23. Suppose f(x) 5 sinx and f(c) = 0.7. What is the value of

2f(−c) + f(c + 2π) + f(c 2 6π)?

24. Suppose f(x) = sinx and g(x) = ln x. Solve ( f ˚ g)(x) = 0.

25. Find the x- and y-intercepts of the parabola whose equation is

(y + 4)² = 4(x + 1).

26. Find the center, foci, vertices, and endpoints of the minor axis of the ellipse whose equation is

x² + 2y² + 2x − 20y + 49 + 0.

27. The slant asymptotes of a hyperbola are y = −5x + 2 and y = 5x − 8. What is the center of the hyperbola?

28. From a point 220 ft from the base of a cell-phone antenna a person measures a 30° angle of inclination from the ground to the top of the antenna. What is the angle of inclination to the top of the antenna if the person moves 100 ft closer to its base?

29. Iodine-131 is radioactive and is used in certain medical procedures. Assume that iodine-131 decays exponentially. If the half-life of I-131 is 8 days, then how much of a 5-gram sample remains at the end of 15 days?

30. The polar coordinate equation r = 3cos4θ is a rose curve with eight petals. Find all radian-measure angles satisfying 0 ≤ θ ≤ 2π for which |r| = 3.

31. Give the three Pythagorean trigonometric identities.

32. Without the aid of a calculator, find the exact value of

cos 80° cos 50° + sin 80° sin 50°.

33. Give the point that is common to the graphs of all exponential functions f(x) = b^x, b > 0, b ≠ 1.

34. Give the y-intercept, the x-intercept, and horizontal asymptote for the graph of f(x) = 4^x − 3.

35. Describe how the graph of y = ln(−x) can be obtained from the graph of y = ln x.

36. Find the asymptotes of the hyperbola

−x² + 10x + 9y² − 54y + 47 = 0.

37. Sketch the graph of the given function.

38. Without doing any work, describe in detail the graph of

39. Solve the linear system

and interpret the solution geometrically.

40. Solve the equation = 0 for x.

41. Here is a nonlinear system of equations taken from a calculus text:

Solve for x, y, and λ.

42. Graph the system of inequalities:

In Problems 43–46, answer the given question about the sequence

128, 64, 32, 16, ….

43. What is the eighth term of the sequence?

44. What is the sum S₈ of the first eight terms of the sequence?

45. Is the sequence convergent or divergent?

46. Does the infinite series

128 + 64 + 32 + 16 + …

have a sum S?

In Problems 47–50, find the nth term a_n of the given sequence.

47. −2, −1, 0, 1, …

48. 0, 3, 8, 15, …

49. 1000, −100, 10, −1, …

50.

51. If d is a digit (any numeral 0 through 9), find a rational number whose decimal representation is 0.ddd ….

52. How many ten-digit telephone numbers are possible within a given three-digit area code if the last seven digits of the telephone number cannot start with 0 or 1?