Download Calculus Pretest

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.
Calculus Pretest
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
1/2
1/3
2. Reduce the expression to lowest terms: x – x
x1/6
3. Simplify: x1/2 – 51/2
x–5
4. 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