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CALCULUS ASSESSMENT REVIEW
DEPARTMENT OF MATHEMATICS
CHRISTOPHER NEWPORT UNIVERSITY
1. Introduction and Topics
The purpose of these notes is to give an idea of what to expect on the Calculus Readiness Assess-
ment for Math 135 and Math 140. It should be stressed that everything contained herein should
essentially be review. While there are some practice problems for each of the major sections, if
you find yourself struggling, you should consider either reviewing precalculus more thoroughly, or
signing up for a lower level course, e.g. Math 110 or Math 130. The problems given here are
similar in style and complexity to those on the assessment, but this should not be treated as a
comprehensive list.
(1) Polynomials and Rational Expressions
• polynomial algebra: addition, subtraction and multiplication
• polynomial factoring
• symbolic fraction computation
• simplifying complex fractions
(2) Equations and Inequalities
• solving linear equations
• solving quadratic equations: factoring and quadratic formula
• solving a word problem: read the problem; introduce a variable and state what the
variable represents; establish an equation and solve the equation.
• |x| = k ⇔ x = korx = −k(two solutions!)
• solving an inequality: change into the standard form; solve the equation and use the
zeros to set up the intervals; check the sign in each interval and select the correct
answer.
(3) Basic Analytical Geometry
• coordinate system
• formulas for midpoint, slope and distance
• equation of a line: point-slope, slope-intercept and general form
1
• graph of a parabola: vertex, x and y-intercepts
2 2 2
• equation of a circle with the center (a, b) and radius r: (x − a) + (y − b) = r
(4) Functions and Their Graphs
• definition of a function
• domain of a function: where the function is well-de?ned
• evaluation of a function
• basic operations and composition of two functions
• computing di!erence quotients √
2 3
• graphs of basic functions: y = ax + b, |x|, x , x , x
• transformations of a basic graph: vertical and horizontal shifts
• inverse function f−1(x): existence and how to find it
• graph of a rational function: using vertical and horizontal asymptotes
(5) Radical Expressions
√ 1/n
• basic operation laws and n a = a
√ √ √ √
• rationalizing the denominator by using the conjugate: ( a + b)( a − b) = a − b
(6) Exponential and Logarithmic Functions
x
• graphs of y = b and y = logb x
• properties of logarithm:
– Product Rule: logb(MN) = logb M + logb N
– Quotient Rule: logb(M/N) = logb M + logb N
– Power Rule: logb Mk == k logb M
– Base Change Formula: logb x = ln x/ ln b
(7) Trigonometry
• degree and radian angle measures
• right triangle trigonometry for an angle 0 < θ < π/2 p
2 2
• coordinate trigonometry for general angles: sin θ = y/r, cos θ = x/r with r = x + y .
• special angle values and unit circle
• graphs of sin x, cos x, tan x, cot x and their variations: period, phase shift, amplitudes
and asymptotes
• inverse trigonometric values
• basic trigonometric identities
• sum, difference and double angle formulas
• solving a triangle: the laws of sines and cosines
2
2. Powers and Roots
Facts to review:
√ 2 √ 2
• For x ≥ 0, x = x; for x < 0, x = |x|.
√
3 3
• For any x, x = x
m n m+n
• x x = x
m n mn
• (x ) = x
−1 1
• x = x
1 √
• xn = n x.
Practice Problems
(1) (−2)3 is equal to
(A) −8 (B) 8 (C) 1 (D) −1 (E) none of these
8 8
(2) p(−4)2 is equal to
(A) −4 (B) 2 (C) 4 (D) −2 (E) none of these
√
(3) 27 is equal to √ √ √ √
� 4
(A) 9 (B) 5+ 2 (C) 3 3 (D) 9 3 (E) 3
√
(4) 200 = √ √ √
(A) 10 2 (B) 20 (C) 10 20 (D) 20 10 (E) none of these
p 8 12
(5) 50x y = √ √
4 6 8 12 4 6 6 10 4 6
(A) 25x y (B) 25x y (C) 5 2x y (D) 5 2x y (E) 5x y
3
3. Lines, Linear and Quadratic Equations
Facts to review:
• Slope-intercept form of a line: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form of an equation: y − y = m(x − x ) where m is the slope and (x ,y ) is a
0 0 0 0
point on the line.
• To solve ax + b = cx + d, isolate x on one side of the equation.
2
• To solve ax + bx + c = 0, either factor (if possible) or use the quadratic equation:
√ 2
−b ± b − 4ac
x = 2a .
Practice Problems
(1) The line with equation 3x − 5y = 15 has slope
(A) −3 (B) 5 (C) −5 (D) 3 (E) −3
3 3 5 5
(2) A line through (2, −5) and (6, −1) has slope
(A) −6 (B) 1 (C) 6 (D) −1 (E) none of these
4 4
(3) A horizontal line through (4, 5) has equation
(A) y = 4 (B) x = 4 (C) y = 5 (D) x = 5 (E) none of these
(4) A line through (1, 6) and (7, 2) has equation
1 2 20 2 20 2 15 2 15
(A) y = x + 7(B) y = − x + (C) y = x + (D) y = − x + (E) y = x +
2 3 3 3 3 3 3 3 3
(5) Find the equation of the line with slope 3/5 and y-intercept (0, −5).
(A) y = 3 x − 3 (B) y = 5 x − 3 (C) y = 3 x + 5 (D) y = 3 x − 5 (E) none of these
5 3 5 5
(6) Solve: x2 + 5x − 6 = 0.
(A) 1, 6 (B) −1, 6 (C) 2, 3 (D) −2, −3 (E) 1, −6
(7) Solve: x2 + 6x − 5 = 0.
√ √ √ √
(A) 6± 56 (B) 6± 16 (C) −6± 56 (D) (−5, −1) (E) −6± 26
2 2 2 2
(8) Let k be a constant. The solution of the equation 2x +7 = kx − 4 is
(A) 4 (B) 2x−11 (C) kx−11 (D) 11 (E) none of these
k−2 k 2 k−2
4
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