Calculus Pretest

I. Linear Equations

1. Find the point-slope form of the equation of the line passing through the points (-1, 2) and (1, -2).

II. Solving Equations/Finding Zeros/Roots of Functions/ Factoring/

(Including Using Quadratic Equation)

  1. Factor by grouping: 5cos2(x) – 5sin2(x) + sin(x) + cos(x)
  2. Factor over the set of real numbers: x2 + 5x + 5.
  3. Factor: x(x + 2)1/2 + (x + 2)-1/2
  4. Factor: x3 – x + 2x2 – 2
  5. Factor: x3 – 8
  6. x3+2x2 – x – 2 = 0
  7. x2 + x – 4 = 0
  8. (3x2 – 8x – 3)/(x + 4) = 0
  9. Solve the system for x and y: 2x – y = 0 and x2 – y = -1
  10. (x + 1)1/2 + (x – 1)1/2 = 2

III. Families of Functions (Domain, Range, Asymptotes, Intercepts, Max, Min, Etc)

  1. Given the graph of f(x) = log4(x), determine the domain, range, asymptotes and intercepts.
  2. Given the graph of g(x) = cos(2x + π /3), determine the domain, range, period, frequency, amplitude, and phase shift.

IV. Rules of Logs and Exponentials

1. logx5 = 2

2. 3x+1 = 15

3. log(x-2) + log(2x-3) = 2log(x)

4. -14 + 3ex = 11

5. 5 + 2ln(x) = 4

6. (32)1/5(9)-1/2

7. If 55 is approximately equal to 3000, then approximate the value of 510?

V. Algebraic Substitution

1. If f(x) = (4x + 2)/(x + 4), then find f(b + 1)

2. Given f(x) = 1/x2, find the difference quotient, [f(x + h) – f(x)]/ h, and simplify completely

3. Find the distance between the following two points: (-1,5) and (3,2).

VI. Solve Inequalities/Polynomial/Absolute Value

  1. |x – 2| < |x + 1|
  2. x2 – 25 > 0

VII. Composition of Functions/Inverse Functions

  1. If f(x) = x2 – x + 1, and g(x) = 2x + 3, find f(g(x))
  2. If r(x) = x2 – 4 and s(x) = x1/2, find s(r(x)) and specify its domain.
  3. Given the graph of an inverse function f-1, sketch a graph of the function f.

VIII. Simplifying Rational Expressions/Rationalizing Algebraic Expressions

  1. Reduce the expression to lowest terms: (x + 1)3(x – 2) + 3(x + 1)2

(x + 1)4

2. Reduce the expression to lowest terms: x1/2 – x1/3

x1/6

3. Simplify: x1/2 – 51/2

x – 5

  1. Simplify: x4 + x – 2

x - 1

IX. Represent Geometric Parameters with Algebraic Expressions

  1. The length of a rectangle is 3 meters more than twice its width. What is the width if the perimeter of the rectangle is 186 meters?
  2. A square picture is mounted on a larger rectangular sheet of poster paper leaving a border around the picture of 2 inches on the bottom and 1 inch on each of the remaining sides. The sheet of poster paper has an area of 42 square inches. If x represents the length of a side of the square picture, then write the equation that can be used to determine the value of x and solve it.
  3. Calculate the surface area and volume of a sphere with radius 2.
  4. Calculate the surface area and volume of a right circular cylinder with radius 3.

X. Trigonometric Functions and Solving Trig Equations (radian to degrees)

1. Prove: csc2(x) – cot2(x) = 1

2. If f(x) = cos(3x), then find f(π/6).

3. For which values of x is sec(x) not defined?

4. Simplify the following expression: sec(θ)cot(θ)sin2(θ)

5. Given that cos(θ) = ½, and θ terminates in quadrant I, evaluate all six trigonometric functions of θ.

6. Solve the following equation for x: 2sin(x) – 1 = 0.

7. Solve the following equation for x: sin(x) + 21/2 = -sin(x)

XI. Language of Mathematics