Math 4 Review Problems


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1 Topics for Review #1 Functions function concept [section 1. of textbook] function representations: graph, table, f(x) formula domain and range Vertical Line Test (for whether a graph is a function) evaluating functions and solving equations using graphs (on paper and on calculator) function properties [1.] increasing, decreasing, constant local maxima and minima, absolute maxima and minima even and odd symmetry (recognizing graphically and confirming algebraically) drawing graphs with specified properties piecewisedefined functions [1.3] function operations [1.4] addition, subtraction, multiplication, and division composition inverse functions [1.5] finding an inverse by reversing the steps of a rule Horizontal Line Test (for whether a graphed function has an inverse) finding an inverse by exchanging the x and y variables of a graph or table finding an inverse graphically by reflecting across the line y = x finding an inverse algebraically transformations, graphs and functions: translations, dilations and reflections [1.6] Polynomials linear functions and quadratic functions in various forms [.1] vertex of a quadratic function (including completingthesquare into vertex form) [.1] graphs of polynomial function (end behavior, multiplicity of zeros) [.3] division concept (relationships between dividend, divisor, quotient and remainder) [.4] polynomial algebraic calculations (especially factoring and dividing) [.4] Remainder Theorem and Factor Theorem (proving and using) [.4] finding zeros and factorizations of polynomials [.3 for real,.5 for complex] using theorems about complex zeros: Fundamental Theorem of Algebra, Complex Conjugates Theorem, odd degree theorem [.5] application/modeling problems [in various sections, and more on handout problem sets] Rational Functions Graphs of rational functions, including all critical features asymptotes, holes and general shape [.6] Limits to describe end behavior and behavior at discontinuities [.6] Equation solving [.7] Application problems [.7] Sign analysis and sign charts [.8]
2 Problems 1. Consider the following functions: f ( x) 3 tan x + sin x = g x ( ) = x h( x) = x + 5 a. Find ( h o g) ( x) and ( g o h) ( x) and their domains. b. Is f(x) even or odd? Prove it.. Describe a sequence of transformations to turn the graph of f ( x) into the graph of g( x). a. f ( x) = x +1, g( x) = ( x + 7) + 5 b. f ( x) = sin x, g ( x ) = 3sin # x "! $ & 4 % 3. Suppose g( x) = x 9 x 5. a. Find g 1 ( x). b. Write g(x) in the form: g(x) = a x h + k c. Describe the transformation of g(x) from f (x) = 1 x. d. Describe the zeros and discontinuities(s) of g(x). e. What are the domain and range of g(x)? f. Sketch a graph of g(x). 4. Find a cubic polynomial, C(x), with real coefficients has the following properties: 1 i is a complex root. the graph of C(x) has an xintercept at x =. C(1) = 1
3 5. All of the zeros are visible on the graph of this polynomial function, and there are no multiplicities higher than. a. Find a function formula for this polynomial graph. b. Without doing the division, find the remainder when this polynomial is divided by (x + ) 6. Solve these equations. Find all solutions. 3 a. x = x + 3x + 4 b. sinθ = 3 7. Consider the following function: g( x) = (x 5x 14) x 4 a. Find the discontinuities of g(x) and identify whether each is an asymptote or a hole. b. Write limit statements describing the end behavior of g(x). c. Write limit statements describing the discontinuities of g(x). d. With the help of your calculator, sketch a graph of g(x).
4 Topics for Review # Exponential and logarithmic functions Exponential functions and their graphs [3.1] Modeling growth and decay using exponential functions [3.] Meaning of logarithms, including exponential and logarithmic functions as inverses [3.3] Logarithmic functions and their graphs [3.3] Solving exponential equations using logarithms [3.5] Arithmetic properties of logarithms (3 properties proved from 3 exponent properties) [3.4] Changeofbase for exponentials and logarithms [3.4] Trigonometric Functions Measures of angles, arcs, and sectors [4.1] Definitions of sin, cos, tan, cot, sec, csc right triangle definition for acute angles [4.] circular definition for all angles [4.3] Trigonometric functions and their graphs sinusoids (sin and cos, possibly with translations and dilations) [4.4] tan, cot, sec, and csc [4.5] inverse trigonometric functions [4.7] Finding trigonometric function values, exactly or approximately [4.4.3] Solving trigonometric equations [4.4, /11 handout] Sinusoidal modeling [4.4, 4.8, /1 handout] amplitude, vertical shift, period, and phase shift realworld applications Geometric applications [4., 4.8, /1/13 handouts] The Law of Sines (including ambiguous case) The Law Cosines Problems 8. a. Write a function formula for a sinusoidal function f(x) having the following properties: Two consecutive maximum points of f(x) are located at (3, 5) and (7, 5). The graph of f(x) is tangent to the xaxis. b. Suppose that the graph of g(x) is formed by shrinking f(x) horizontally by a factor of 8. Write a function formula for g(x). 9. For triangle RST, given that R = 45, RS = 8, and ST = 10, find the remaining measurements of the triangle. There may be one or two sets of answers.
5 10. For quadrilateral WXYZ, given that XY = 10, Y = 50, YZ = 8, Z = 110, and ZW = 6: a. Find the remaining sides and angles. b. Find the area. 11. Suppose that the depth of the water in a harbor is 0 feet at low tide, 30 feet at high tide, and fluctuates in such a way that it can be modeled with a sinusoidal function. Let t represent time measured in hours, and let D(t) represent the water s depth in feet at time t. Suppose that a low tide occurs at time t = 5.4 and the next high tide occurs at time t = a. Using the sine function, write a function formula for D(t). b. Find all times t at which the depth of the water is 8 feet. (Use decimal approximations accurate to the nearest 0.01.) 1. The halflife of radium is about 1600 years. Suppose that an archeological sample is found to contain 00 milligrams of radium. How many milligrams of radium were there 5,000 years ago? 13. Let a = log 3 and b = log 3 5. Express log in terms of a and/or b.! 14. Given w = bx $ # &, write log " y 3 b w in terms of log b x and log b y. % Prove that log b c =. (You can use any formulas proved in class.) log b c " 16. Find an equation for an exponential function passing through the points 1, % $ ' and,18 # 3& ( ). 17. An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person s system decreases by about 9%. How much ibuprofen is left after 6 hours? How long will it take for the amount of ibuprofen to decrease to 10% of its original value?
6 Topics for Review #3 Trigonometric Identities [5.15.] angle sum(difference) identities double angle identities solving and proving trigonometric equations with identities Vectors [6.16.] vector concept: runandrise or directionandmagnitude equivalent arrows representing the same vector vector operations (addition, subtraction, scalar multiplication, dot product) using both algebraic and visual approaches vector magnitude (length) and unit vectors parallel and perpendicular vectors; angle between vectors geometric proofs using vectors Parametric equations [6.3] parametric equation graphing, by hand and on calculator writing parametric equations for lines, line segments, and circles eliminating the parameter t modeling D motion using parametric equations (incl. gravity problems) Polar coordinates [6.4] polar coordinate concept converting coordinates between rectangular and polar converting equations between rectangular and polar polar equation graphing, by hand and on calculator application problems involving polar graphing and distances Complex numbers in polar form [6.6] writing complex numbers in polar form (trigonometric form) polar form multiplication, division, and powers (DeMoivre s Theorem) The material about linear system and matrices from Chapter 7 has not been included in the review because we just finished learning it, but could appear on the exam. Problems Prove the identity + = sec x. Present your proof as a chain of 1+ sin x 1 sin x 1 1 equal expressions that begins with + and ends with sec x. 1+ sin x 1 sin x 19. Derive a formula for cos(u) that involves sin u but does not involve cos u. " 0. Prove that cos(θ) = sin$ π # θ % ' using sin(α β). &
7 1. A woman has a collection of bills in her wallet. Each bill is worth $1, $5 or $10. She has 19 bills, amounting to $9. There are twice as many ones as tens. What bills will you find in her wallet?. At Fenway Park, the Green Monster is 37 feet high and is located 310 feet from home plate. Suppose a batter hits a ball at an elevation of 4 feet from the ground at an initial velocity of 11 ft/s at an angle of 8º. Will a spectator seated atop the Green Monster be able to catch the ball? 3. Vector v is the directed line segment from (7,1) to (5,8). Vector w has component form <,5>. a. Draw the vector w +.5v on the axes at right. b. Use a vector method to find the angle between w and v. c. Find a vector of magnitude in the direction of v. 4. Let z = 1 i. 8 a. Find z. Express you answer in polar form b. Find z. 5. Fill in the conversion table below: cartesian convert polar x ( 1, + y 3 ) = 1 ( 4, 5π 6 ) r = 4cscθ 6. a. Write a polar equation (in r = L form, if possible) for the line whose rectangular equation is y = x + 3. b. Convert polar equation r = if possible). c. Convert the polar equation r = 4cosθ to rectangular form. 1 into a rectangular equation (in y =L form, cos( θ ) + sin( θ )
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