Practice Exercises For Mathematics Placement Test - Test (Corresponds to Precalculus Competency - Preparedness for M5 The Test Placement exam is a multiple choice exam covering topics typically found in a Precalculus course. Passing the exam means that you are prepared to take M5 - Calculus I. Below are practice exercises to review some of the material required for the exam. Answers to the practice exercises can be found on the last pages. Major Topics: A student taking the exam should be prepared to Factor and simplify polynomial, rational, radical and absolute value expressions. Recognize and graph equations of circles and parabolas (may need completing the square. Solve equations and inequalities involving absolute values, polynomial, and rational expressions. Work with function notation and function operations (including composition and the difference quotient. Work with quadratic and rational functions, as well as inverse functions. Graph polynomial, rational, exponential, and logarithmic functions. Use the Factor Theorem, the Remainder Theorem, and polynomial division (either long division or synthetic division to find integer, rational, and irrational roots of polynomial equations. Solve exponential and logarithmic equations. Solve and graph systems of linear equations and inequalities in two or three variables. Know the unit-circle definition of trigonometric functions. Know right triangle trigonometry (opposite, adjacent, and hypotenuse. Evaluate trigonometric and inverse trigonometric expressions. Change radian measure to degree measure and vice versa. Graph trigonometric functions, including transformations. Know and use fundamental trigonometric identities to simplify expressions. Verify trigonometric identities. Solve trigonometric equations. Revised May 6, 00
. Find all solutions to the following equations. (a x + x 7 = 0 (b x 5x 6 x + = 0 (c 4x + = 7 (d x + = 6. Determine the set of all solutions for each, graphing inequalities on a number line. (a x + < (d (x 5(x + > 0 (b x 7 (e (x + (x x 0 (c (x (x + 0 (f x + > x. Graph and state the coordinates of the vertex of each. (a y = (x 5 + (b y = x x 4. If f(x = x x and g(x = x +, evaluate each of the following. (a f(5 (e g( (i f( + h f( h (b (f + g(x (f (f g( (j g(x + h g(x h (c (fg(x ( f (g ( g (d (f g(x (h (g f(4 5. State the domain and range of each; then graph each. (a f(x = 7x + 4 (b g(x = x + (c h(x = x + 6. Determine the inverse functions of f(x = x. 7. Graph each. (a y x + (b y > x 8. Find the set of solutions for each systems of equations. { x y + z = x y = (a (b x y + z = x + y = 7 x y + z = (c { y = x + y = x + 5 9. Write an equation of the circle with center at the origin and radius of. 0. Determine the center and radius of the circle (x + (y + = 4.. Determine the center and radius of the circle x + y 4y =.. Determine whether x is a factor of x + 4x x + 4. [Use of the Factor Theorem would be a quick way to do this.]. Find the quotient and remainder if x 4 x x + 7 is divided by x +. [Use of synthetic division would be a quick way to do this.] 4. Based on the coefficient of the x 4 term and the constant term, state the candidates for rational roots of x 4 + x 4x + = 0.
5. Find all real solutions (giving exact values of x 4 x x 4x + = 0. 6. Graph each. (a y = x (b y = x (c y = log x 7. Find all solutions for each. (a log 8 = x (d 4 x = x (b log 7 x = 0 (e log(x + log(x = log 4 (c 5 x = 5 8. Graph each. (a f(x = x x + 9. Find the coefficient of x y. (b g(x = x x 4 (a (x + y 4 (b (x y 4 0. Use the unit-circle definition of the trigonometric functions to obtain the values of each, if t is as indicated in the figure to the right. (a sin t (b cos t (c tan t (d cot t (e sec t (f csc t (a, b y t x. If sin θ = 5 and θ is in quadrant III, determine each. (a cos θ (b tan θ (c cot θ (d sec θ (e csc θ. Given θ as indicated in the figure at the right, determine each of the following. (a sin θ (b cos θ (c tan θ (d cot θ (e sec θ (f csc θ θ 5. State the values of the following. ( (a sin (0 (b cos ( 6 ( (f tan (g sin 4 (c tan (0 ( (h cos ( (d sin 4 (i tan (60 (e cos (45
4. Write each in terms of a trigonometric function of an angle in the first quadrant. (a sin (50 (b cos (5 (c tan (00 5. Change each radian measure to degree measure. (a (b (c 7 6 ( + cos. 7 (d 4 (e 6. Find the exact value of sin ( 7 7. Graph each. (a y = sin x (b y = cos x (c y = tan x (d y = cos x (e y = sin x ( 8. Find the period and amplitude of the graph of y = cos x. 4 9. Reduce each to a single function of the argument θ. (a cos θ csc θ (b sec θ sin θ tan θ 0. If sin θ = and cos θ =, find sin θ.. If tan φ = 4, find cos φ.. Find the set of solutions in the interval 0 x for each equation. (a sin x = (d sin x + = 0 (b cos x + = (e cos x sin x = (c tan x = (f tan x = 0. Let C be the right angle of triangle ABC. (a If the length of the side opposite A is 6 centimeters and the measure of B is 60, find the length of the hypotenuse. (b If the measure of A is 45 and the length of the side opposite B is 5 feet, find the length of the side opposite A. 4. Find the value of each. ( (a sin (b cos(tan
You should know the following identities. Logarithmic Identities Exponent Identities log(ab = log A + log B log ( A = log A log B B log(a n = n log A A m A n = A m+n A m A n = Am n (A m n = A mn (AB n = A n B n ( n A = An B B n Trigonometric Identities sin x + cos x = sin(x = sin x cos x + tan x = sec x cos(x = cos x sin x + cot x = csc x tan x = cot x = sin x cos x sec x = cos x csc x = sin x cos x sin x = tan x sin( x = sin x cos( x = cos x tan( x = tan x
. (a x = ± 7 Answers For Test.54, 4.54 (b x = 6, (c x = 7 (d x = 5, 7. (a 5 < x < (b x or x 8/ (c x 5 0 0 (d x < or x > 5/ 8 0 (e x or x < (f x < or < x < 0 0 5 0 0. (a Vertex (5, (b Vertex (, 4. (a 45 (b x + (c x + 5x x (d (x + (x + = x + x + 5 (e 0 (f 9 (g 4 (h (i 7 + h (j 5. (a Domain: [ 4/,, Range: [0, (b Domain: all real numbers (R, Range: all real numbers (R (c Domain: all real numbers (R, Range: [, 6. f (x = + x
7. (a (b 8. (a x =, y = 5 (b x = 0, y =, z = (c x = 4, y = 7 and x =, y = 9. x + y = 9 0. Center (,, radius. Center (0,, radius 4. Not a factor.. Quotient: x x + x 4, Remainder: 4. ±, ± ±, ± 5. x =,, ± 4 = ± 6. (a (b (c 7. (a x = (b x = (c x = (d x = ln 4 ln ln 4 ln = ln 4 9 ln 64 (e x =
8. (a (b 9. (a 4 (b 08 0. (a b (b a (c b a (d a b (e a (f b. (a 4 5 (b 4 (c 4 (d 5 4 (e 5. (a 5 (b (c 5 (d 5 (e (f 5. (a (b (c (d (e (f (g (h (i 4. (a sin 0 (b cos 45 (c tan 60 5. (a 60 (b 90 (c 0 (d 5 (e 80 6. 7. (a (c (d.5 (b (e.5 8. Amplitude =, Period = 9. (a cot θ (b cos θ 0. 4 9
. 7 5. (a 6, 5 6 (b, 5 (c 4, 5 4 (d 4, 7 4 (e 6, 5 6, (f,, 4, 5. (a cm (b 5 ft 4. (a 6 (b