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 Assessment for Math 15 and Math 10. 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 10. 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 ) 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 = ktwo 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. ) 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 equation of a circle with the center a, b) and radius r: x a) + y b) = r ) 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 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 n basic operation laws and a = a1/n rationalizing the denominator by using the conjugate: a + b) a b) = a b 6) Exponential and Logarithmic Functions graphs of y = b x and y = log b x properties of logarithm: 7) Trigonometry Product Rule: log b MN) = log b M + log b N Quotient Rule: log b M/N) = log b M + log b N Power Rule: log b M k == k log b M Base Change Formula: log b x = ln x/ ln b degree and radian angle measures right triangle trigonometry for an angle 0 < θ < π/ coordinate trigonometry for general angles: sin θ = y/r, cos θ = x/r with r = p 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
. Powers and Roots For x 0, x = x; for x < 0, x = x. For any x, x = x xmx n = xm+n x m ) n 1 1 = x = x x 1 n x n = x. Practice Problems mn 1) ) is equal to A) 8 B) 8 C) 1 8 D) 1 8 E) none of these ) p ) is equal to A) B) C) D) E) none of these ) 7 is equal to A) 9 B) 5 + C) D) 9 E) ) 00 = A) 10 B) 0 C) 10 0 D) 0 10 E) none of these 5) p 50x 8 y 1 = A) 5xy 6 B) 5x8y 1 C) 5 xy 6 D) 5 x6y 10 E) 5xy6
. Lines, Linear and Quadratic Equations 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 0 = mx x 0 ) where m is the slope and x 0, y 0 ) is a point on the line. To solve ax + b = cx + d, isolate x on one side of the equation. To solve ax + bx + c = 0, either factor if possible) or use the quadratic equation: b ± b ac x =. a Practice Problems 1) The line with equation x 5y = 15 has slope 5 5 A) B) C) D) E) 5 5 ) A line through, 5) and 6, 1) has slope 6 A) 6 B) 1 C) D) 1 E) none of these ) A horizontal line through, 5) has equation A) y = B) x = C) y = 5 D) x = 5 E) none of these ) A line through 1, 6) and 7, ) has equation 1 0 0 15 15 A) y = x + 7B) y = x + C) y = x + D) y = x + E) y = x + 5) Find the equation of the line with slope /5 and y-intercept 0, 5). 5 A) y = x B) y = x C) y = x + 5 D) y = x 5 E) none of these 5 5 5 6) Solve: x + 5x 6 = 0. A) 1, 6 B) 1, 6 C), D), E) 1, 6 7) Solve: x + 6x 5 = 0. A) B) C) D) 5, 1) E) 6± 56 6± 16 6± 56 6± 6 8) Let k be a constant. The solution of the equation x + 7 = kx is kx 11 11 A) B) x 11 C) D) E) none of these k k k
. Functions Function composition, f gx) = fgx)). Inverse function: gx) = f 1 x) if and only if fgx)) = x and gfx)) = x. Practice Problems 1) If fx) = 10 x and gx) = x x, find the value of fg )). A) B) C) 98 D) E) ) If fx) = x 1, find the value of x such that ffx)) = 9. A) B) C) D) 5 E) 6 ) If fx) = x 9 and fx) = 9, what is x? A) 0 B) 1 C) 1 D) 9 E) 9 ) If fx) = x + 1 and gx) = x + 1), then fgx)) = A) x + x + 1 B)x + x + C) x + x + x + x + 1 D) x + x + 6x + x + E) none of these 5) If fx) = x + 1 and gx) = x + 1), then gfx)) = A) x + x + 1 B)x + x + C) x + x + x + x + 1 D) x + x + 6x + x + E) none of these fx+h) fx) 6) If fx) = x, then = h A) 0 B) h C) x D) x + h E) xh 7) If fx) = x + 5, then f 1 x) = 1 A) B) 1 x + 5 C) 1 x 5 D) 1 x 5 E) none of these x+5 8) If fx) = x 1, then f 1 x) is 1 A) x + 1 B) x C) x + 1 D) x 1 E) none of these 1 5
5. Polynomial and Rational Functions A polynomial is a function of the form px) = a n x + a n 1 x of px) is n, the highest power of x. Rational functions are functions of the form Rx) = p q n x) x) n 1 + a 1 x + a 0. The degree where px) and qx) are polynomials. px) rx) px)sx) + rx)qx) To add rational functions, use a common denominator: + =, qx) sx) sx)qx) possibly simplifying when done. px) rx) px)rx) To multiply, multiply numerators and denominators: =. qx) sx) qx)sx) A rational function, Rx) = px)/qx) has a horizontal asymptote if degree px) degree qx). A rational function may have a vertical asymptote where qx) = 0. Practice Problems 1) Expand: x )x + )x 1). x x A) x + x B) x + x 1 C) x + 7x + x 6 D)x + 6 E) x + x 7x x + 6 ) Find all real zeros of the polynomial px) = x ) x + 5)x + 1). A) x = B)x =, 5 C)x = 5 D) x =, 5 E) x =, 5, 1 x ) Combine: x 1 x x x+ x x x x+1 x x 1 A) B) C) D) E) x 1 x 1)x x 1)x xx 1) xx 1) x ) Combine: x + x 1 +x 1 x x x A) x 1 B) x 1 C) 0 D) x 1 E) x 1 5) Solve: x = 5 x+6 15 15 A) B) 15 C) 15 D) E) 10 6
x 6) Solve: 5 = 1 x 10 7 5 A) 1 B) C) 6 D) E) 6 8+ 7) Simplify: 1 x + 1 x x+1 8x+1 1 A) x + 1 B) C) D) E) + x+1 x+1 x x 1 8) Let Rx) =. What is the horizontal asymptote of Rx)? x + 1 A) y = 1 B) y = 0 C) x = 1 D) x = 0 E) none of these x 1 9) Let Rx) =. What is the vertical asymptote of Rx)? x + 1 A) y = 1 B) y = 0 C) x = 1 D) x = 0 E) none of these x 1 10) Let Sx) =. What is the horizontal asymptote of Sx)? x + 1 1 1 A) x = B) y = C) y = D) x = 0 E) none of these 11) If fx) = x+1, its graph will have x )x 1) A) one horizontal and three vertical asymptotes B) one horizontal and two vertical asymptotes C) one horizontal and one vertical asymptotes D) zero horizontal and one vertical asymptotes E) zero horizontal and two vertical asymptotes 7
6. Geometry Pythagorean Theorem c = a + b Area of a rectangle, length width Area of a triangle, 1 base height Area of a circle, πradius) Circumference of a circle, πr Distance between two points, p x x 1 ) + y y 1 ) Practice Problems 1) The diagonal of a square is 7. The side is A) 6 B) C) 6 D) E) none of these ) Numerically, the area of a circle is four times its circumference. Its radius is A) B) π C) 8 D) 8 E) 8π ) The length of a rectangle is twice its width. Numerically, its area is equal to the length of its diagonal. Its width is 5 5 A) B) 1 C) D) E) ) What is the distance between, ) and, 1)? A) 10 B) 10 C) 50 D) 50 E) none of these 5) What is the distance between, 1) and, )? A) 6 B) 9 C) D) 17 E) none of these 8
7. Inequalities Open and closed intervals x a has solution set [ a, a]; x < a has solution set a, a). x a has solution set, a] [a, ); x > a has solution set, a) a, ) Polynomials can only change sign at a root. Rational functions can only change sign at a root or at an asymptote. Practice Problems 1) Solve: x + 5 x. A), 9] B) [9, ) C) [ 9, 9] D) 9, 9) E) none of these ) Solve: x 5x + > 0. A), 1), ) B) 1, ) C), 1] [, ) D) [1, ] E) none of these x + 1 ) Solve: <. x + A) 5, ) B), 5), ) C) 5, ) D), 5) E) none of these ) Solve: x 5. A) [ 1, 1] B) [ 9, 9] C) [ 1, 9] D) [ 1, 9] E)[ 9, 1] 5) For x 7, which of the following is not a solution? A) B) 10 C) 8 D) 10 E) 6) Solve: x 5 A) [7, ) B), ] [7, ) C), ] D) [, 7] E), ) 7, ) 9
8. Exponentials and Logs If b > 1, then b x 0 as x and b x as x. If 0 < b < 1, then b x as x and b x 0 as x. log b x) is the inverse of b x ; log b b x ) = x and b log b x) = x. log b xy) = log b x) + log b y) log b x ) = log b x) log b y) y log b x y ) = y log b x) Practice Problems 1) What is the exact value of log 1 )? 8 A) 8 B) C) 7 D) E) a b ) If log a) =, log b) = 1, and log c) = then log is c A) 5 B) C) 7 D) E) 11 ) Solve: 9 x = 7 A) x = B) x = C) x = 9 1 D) x = E) none of these ) Solve: 5 x+ = x 1 log)+ log5) log) log5) log5) log5) log5) log5) A) B) C) D) E) none of these log) log5) log)+ log5) log)+ log) log5)+ log5) 5) Solve: log x + logx 15) =. A) 5 B) -5 C) -5, 0 D) 0 E) -1 10
9. Trigonometry Unit circle angle in degrees and radians, its cosine and its sine. If x, y) is the point on the circle at the end of a radius forming an angle θ with the positive x-axis, then x = cos θ and y = sin θ. Pythagorean identities: sin θ + cos θ = 1 1 + cot θ = csc θ tan θ + 1 = sec θ Sum and Difference Identities sinα ± β) = sin α cos β ± sin β cos α cosα ± β) = cos α cos β sin α sin β Double angle Identities sin θ = sin θ cos θ cos θ = cos θ sin θ Arc cosine: cos 1 x) is the angle θ, 0 θ π with cosθ) = x. Arc sine: sin 1 x) is the angle θ, π π θ with sinθ) = x. Arc tangent: tan 1 x) is the angle θ, π < θ < π with tanθ) = x. Graphs of y = A sinbx + C) + D and y = A cosbx + C) + D 11
y 1, ) 0, 1) 1, ) ), 1, ) 5π 6 π π 10 π 90 60 π π π 6, ), 1 ) 150 0 1, 0) 1, 0) π 180 60 0 π x ), 1 7π 6, ) 5π 10 1, ) π 0 70 00 π 0, 1) 5π 0 7π 11π 6 1, ),, 1 ) ) 1
Practice Problems 1) The exact value of cosπ/) is: A) B) 1 C) D) E) ) The exact value of sin5π/) is A) B) 1 C) D) E) ) The exact value of tan π/) A) 1 B) 1 C) D) E) 1 ) If θ is in the first quadrant and cosθ) = 1, then sinθ) is A) B) C) D) E) none of these 5) If sin α = and cos α > 0, then sin α is: 5 5 A) B) C) D) E) none of these 5 9 6) The exact value of sin 1 1 ) is: A) π B) π C) π D) E) none of these 6 7) The exact value of cos 1 1 ) is: A) π B) π C) π D) E) none of these 6 8) sin x tan x + cos x = A)tan x B) csc x C) sec x D) csc x E) sin x cos x sin x 9) = 1 tan x A) cos x B) sin x C) sec x D) csc x E) sin x 1 1 10) 1+cos θ + 1 cos θ = A) cos θ B) sin θ C) tan θ D) csc θ E) cot θ 1
11) Find the exact value of sin 0 sin 5 cos 5. A) 0 B) 1/ C) 1 D) E) no solution 1) The period and phase shift of fx) = cosx + π/) is A) π/, π/6 B) π/, π/6 C) π/, π/ D) π, π/ E) π/, π/ 1) If sin α = 1, cos β > 0, sin β = 1/, then sinα + β) is A) 1/ B) / C) / D) / E) / 1) The graph below best represents which function? 1 A) sinx) B) sinx) C) cosx/) D) cos x E) cosx/) 1
10. Answers to Practice Problems Section 1) A Section 6 1) A ) C ) C ) C ) D ) A ) A 5) C 5) C Section 1) D Section 7 1) B ) B ) A ) C ) D ) B ) C 5) D 5) D 6) E 6) B 7) C Section 8 1) E 8) D ) A Section 1) D ) A ) B ) A ) D 5) D ) D Section 9 1) A 5) B ) E 6) D ) C 7) C ) E 8) C 5) D Section 5 1) E 6) B ) B 7) A ) A 8) C ) E 9) A 5) B 10) D 6) C 11) A 7) C 1) A 8) B 1) E 9) E 1) B 10) B 11) B 15