Review for Cumulative Test We will have our second course-wide cumulative test on Tuesday February 9 th or Wednesday February 10 th, covering from the beginning of the course up to section 4.3 in our textbook. This packet has a review topic outline, review problems, and answers. Topic outline 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 piecewise-defined 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 [.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 [.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] 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] Change-of-base for exponentials and logarithms [3.4] Trigonometric Functions Angles and their measures: degrees and radians, arc length and sector area [4.1] Trigonometric functions of acute angles right triangle trig [4.] Trigonometry extended: the circular functions: trig functions of any angle, periodic functions, the unit circle [4.3] Review problems Functions 1. Describe a sequence of transformations to turn the graph of f ( x) into the graph of g( x). a. f ( x) = x 3 1, g( x) = ( x + 7) 3 b. f (x) = x, g(x) = 3 x+4 +1. Let f ( x) = x + 5 and g( x) = x. Find ( g! f )( x) and ( g)( x) 3. Suppose g( x) = x + 3 x 5. a. Find g 1 ( x) b. Find the range of g x f!. ( ). c. Find lim g( x). 4. Decide whether each function is odd, even, both, or neither. Then prove it. f x = x a. ( ) 4 x b. f ( x) = x 3 + 5
Polynomials 5. A graph of the function f ( x) = x 4 + 6x 3 +10x + 6x + 9 is shown to the right. a. Factor f(x) as much as possible in the real number system. b. Factor f(x) as much as possible in the complex number system. 6. Let f ( x) = x 5 + x 3 Rational Functions a. Without using a calculator, use limit statements to describe the left- and right-end behavior of f(x). b. What is the remainder when f(x) is divided by (x + )? Is (x + ) a factor of f(x)? c. Find the real zeros of f(x). You may use your calculator. 7. Consider the function: f ( x) = x 3 x + x 1 a. What is the domain of f(x)? b. Does f(x) have any holes? If so, find the x and y coordinates of the hole(s). c. Does f(x) have vertical asymptotes? If so, write the equation(s) of the asymptote(s). d. Describe the end behavior of f(x) using limit statements. e. Use limit statements to describe the behavior of f(x) at the discontinuities you identified in parts b and c. 8. Consider a rational function, f(x), with the following properties: Vertical asymptotes at: x = 3 and x = 4 ( ) = and lim ( ) = lim f x f x x f( 1) = 0 and f( 6) = 0. a. Find a possible function formula for f(x). b. Sketch a graph of f(x). Exponential and Logarithmic Functions 9. Suppose f x ( ) = a b x where f (0) = 3 and f (3) = 3/64. a. Find a function formula for f(x). b. If x is time measured in minutes, by what percentage (%) does f decrease each minute?
10. Strontium-90 is a radioactive substance that decays exponentially according to the equation A( x) = A 0 e.044 x, where x is the time in years, A(x) is the amount present, and A 0 is the amount present initially (when time = 0). To answer these questions, assume that that when time = 0, there are 400 grams of Sr-90 present. a. How much Sr-90 will be left after 1 years? b. When will 100 grams be left? (Solve algebraically.) c. What is the half-life of Sr-90? d. Find A 1 ( x). e. What is A 1 ( 50)? Explain the meaning of your answer in the context of the problem. 11. Without using your calculator, find the exact values: a. 5 log 5 6 b. log 5 45 log 5 3 c. log 5 9 ( ) log 5 1 3 Trigonometry 1. The point ( 1, 1) is on the terminal side of an angle, θ, in standard position. a. Give the smallest positive angle measure for θ in both degrees and radians. b. Find the six trig functions of θ 1. Evaluate the following without using your calculator. a. cot 3π ( ). b. sin# 11π! " 6 $ % & c. sec " $ π # 3 13. Directions: Find tanθ and cscθ given each set of information below: % ' & a. cosθ = 5 7, tanθ < 0 b. cotθ is undefined and cosθ > 0 Answers 1. a. The left shift can be done in any order, but the order matters in the vertical direction. 1. Shift left by 7 units.. Shift up 1. 3. Reflect over the x axis. b. 1. Left by 4.. Stretch vertically by a factor of 3 3. Up by 1. ( g o f )( x) = g( f ( x)) = g( x + 5) = x + 5 ( f o g)( x) = f ( g( x)) = f ( x) = ( x) + 5 = x + 5 3. a. g 1 ( x) = 5x+3 x b. Since the range of g(x) is the same as the domain of g 1 x (, ) (, ) ( ), the range of g(x) is:
c. One method: g(x) has a horizontal asymptote at y =, so lim g( x) =. Another method: g -1 (x) has a vertical asymptote at x =, so g 1 (x) has a horiz. asymptote at y =. 4. a. f ( x) = ( x) 4 ( x) = x 4 x = f ( x ). f ( x) = f ( x), so f is even. b. ( x) = ( x) + 5 = x + 5 3 f. f ( x) f ( x), f ( x) f ( x), so f is neither even nor odd. 3 5. f ( x) = x 4 + 6x 3 +10x + 6x + 9. From the graph, we can see that x+3 is a factor with multiplicity =. Using long division: a. f (x) = (x + 3) (x +1) b. f (x) = (x + 3) (x + i)(x i) 6. f ( x) = x 5 + x 3 7. f ( x) = a. End behavior is Up/Down: lim f ( x) = and lim f ( x) = x b. The remainder is: f ( ) = 37. Therefore, x+ is not a factor. c. One real zero: x = 1. x 3 x + x 1 a. Domain: (, 4) ( 4,3) (3, ) b. Hole at: (3, 71 ) c. Vertical asymptote at x = 4. d. lim f ( x) = 0 and lim f (x) = 0 x 8. 9. e. lim x 4 a. f (x) = f (x) =, lim x 4 + (x +1)(x + 6) (x 3)(x + 4) f (x) =, lim f (x) = 1 x 3 7 b. Check your graph with your calculator.! a. f (x) = 3 1 $ # & " 4 % x! b. f (x) = 3 1 $ # & " 4 % x = 3 (.5) x, so decreases by 75% each minute.
10. A( x) = 400e.044 x a. A(1) = 98.5 grams b. x = ln(1/ 4) = 56.8 years.044 c. 8.41 years. d. A 1 (x) = 1.044 ln " x % $ ' # 400 & e. A 1 (50) = 85.. In 85. years, 50 of the original 400 grams of Sr-90 will be left. 11. a. 36 b. 1 c. 1. The point is: ( 1,1) a. θ = 135 o or 3π/4 radians b. sinθ = y r = 1 cscθ = r y = cosθ = x r = 1 tanθ = y x = 1 secθ = r x = cotθ = x y = 1 13. a. Undefined b. 1 c. 14. a. tanθ = 4 5, cscθ = 7 4 b. tanθ = 0, cscθ is undefined