Math 111 Sample Final Exam Winter 2011 Name ANSWERS (150 points) Date Please show ALL of your work if full or partial credit is desired. Communicating your solution is as important as stating your answer. Do all graphs neatly on graph paper; be sure to label your graphs and axes. Where ever possible find exact answers, otherwise to the nearest thousandth. 1. Given f (x) = (a) f () = (b) f (g(2)) (c) f (g(x)) x+2 and g(x) = 2x + 5, find: x−2 +2 −2 = 11 7 (2x+5)+2 = (2x+5)−2 (d) g(f (x)) = 2 (e) f (f (x)) = x+2 x−2 = 2x+7 2x+3 +5= x+2 ( x−2 )+2 = x+2 ( x−2 )−2 7x−6 x−2 3x−2 −x+6 2. Given f (x) = 2x − 3 and g(x) = (a) f −1 (x) = (b) g −1 (x) = x+3 2 2 + x 2 , find: x−2 2 3. Solve each equation: (a) 2x = 16 x = log2 (16) = 4 (b) log3 (x) = 3 (c) logb (16) = 2 x = 33 = 27 b2 = 16 =⇒ b = √ 16 = 4 (d) 2x = 5 x = log2 (5) ≈ 2.32192809 4. Given f (x) = 2x and g(x) = −2x−1 + 2 (a) Find the domain of each function. Df : (−∞, ∞); Dg : (−∞, ∞) (b) State the asyptote(s) of each function. yf = 0; 1 yg = 2 (c) Graph each function. (d) State the range of each function. Rf : (0, ∞); Rg : (−∞, 2) (e) Find the inverse of each function. f −1 (x) = log2 (x); g −1 (x) = log2 − (x − 2) + 1 5. Given f (x) = log3 (x) and g(x) = − log3 (x − 1) + 2 (a) Find the domain of each function. Df : (0, ∞); Dg : (1, ∞) (b) State the asyptote(s) of each function. xf = 0; (c) Graph each function. 2 xg = 1 (d) State the range of each function. Rf : (−∞, ∞); Rg : (−∞, ∞) (e) Find the inverse of each function. f −1 (x) = 3x ; g −1 (x) = 3 − (x−2) +1 6. To what expontent would you raise 2, to get 12? The exponent is log2 (12) ≈ 3.584962500721156181453738943947816508759814407692481060455752654 7. Analyze the graph of the function f (x) = 2x3 − 11x2 + 12x + 9 (a) Determine the end behavior of the graph of the function. On the right, as x −→ ∞, y −→ ∞ On the left, as x −→ −∞, y −→ −∞ (b) Find the x- and y-intercepts of the graph of the function. 2 f (x) = 2 x + 21 x − 3 y-intercept (0, 9), x-intercepts (0, −1/2), (0, 3) (c) Determine the zeros of the function and their multiplicity. Does the graph cross or touch at the x-axis at each x-intercept? x = − 12 , multiplicity 1, crosses x = 3, multiplicity 2, touches (d) Does the function have any local or absolute maximums? If so, at what x-value and what is the maximum? Local maximum of about 12.703 when x = 32 (e) Does the function have any local or absolute minimums? If so, at what x-value and what is the minimum? Local minimum of 0, when x = 3 (f) On what interval(s) is the function increasing? (−∞, 32 ), (3, ∞) (g) On what interval(s) is the function decreasing? ( 32 , 3) (h) Graph the function. Include at least 5 points. 3 8. r(x) = x2 − 3x − 18 x−2 (a) State the domain. (−∞, 2) ∪ (2, ∞) (b) State the vertical asymptotes, if any. x=2 (c) State the horizontal asymptotes, if any. NONE (d) State the oblique asymptotes, if any. x −1 2 x−2 x − 3x − 18 − x2 + 2x − x − 18 x −2 − 20 4 So the oblique asymptote is y = x − 1 (e) State the intercepts. y-intercept (0, 9), x-intercepts (0, −3), (0, 6) (f) Graph the function. (g) State the range of the function. All Real numbers 9. Graph f (x) = (x + 2)2 − 3. Label the axis of symmetry, vertex, the x- and y-intercepts. Draw the axis of symmetry as a dotted line. State the domain and the range. State the x-intercepts exactly and approximately to the nearest hundredth. vertex: (−2, −3) axis: x = −2 y-intercept: (0, 1) √ x-intercepts: (−2 ± 3 , 0) ≈ (−0.27, 0), (−3.73, 0) Domain: (−∞, ∞) Range: [−3, ∞) 5 10. Graph f (x) = x2 − 6x + 3. Label the axis of symmetry, vertex, the x- and y-intercepts. Draw the axis of symmetry as a dotted line. State the domain and the range.State the x-intercepts exactly and approximately to the nearest hundredth. vertex: (3, −6) axis: x = 3 y-intercept: (0, 3) √ x-intercepts: (3 ± 6 , 0) ≈ (5.45, 0), (0.55, 0) Domain: (−∞, ∞) Range: [−6, ∞) 6 11. Graph g(x) = − 12 (x − 2)2 + 3 . Also graph of f (x) = x2 , our basic parabola and describe how the graph of g(x) is related to the graph f (x), in terms of horizontal and vertical function transformations. Compared to f (x) = x2 , g(x) is shifterd right 2, flipped vertically about the x-axis, vertically compressed by 12 , and shifted up 3. 7 √ 12. Graph s(x) = 2x − 3 . State the domain and range. Domain: [1.5, ∞) Range: [0, ∞) 13. The data below was obtained from a random sample of ten full-time students where h is the hours of video games played and G is the student’s cumulative grade-point average. h 0 0 2 3 3 5 8 8 10 12 G 3.49 3.05 3.24 2.82 3.18 2.78 2.31 2.54 2.03 2.51 The line of best fit is G(h) = −0.0942 · h + 3.2763. (a) Write a sentence to interpret the slope in this context. 8 For each additional hour a student spends playing video games, their GPA will drop by 0.0942 points. (b) Find the y-intercept. What does it mean in this situation? The average GPA of a student does not play video games is about 3.2763. (c) Find the x-intercept. What does it mean in this situation? If a student plays video games 24 hours a day, his GPA will be 0. 14. Gary receives an email from an investment firm that promises to double his investment in 5 years if he gives them $5,000 now. What is the effective annual yield of this proposal? 1 r = 2 5 − 1 ≈ 0.148698355 15. A colony of bacteria grows exponentially according to the function B(t) = 100e0.045t where B is measured in grams and t is measured in days. (a) Determine the initial population of the bacteria colony. Initially there are 100 g of bacteria. (b) What will the population be after 5 days? In 5 days there will be 125.232 g of bacteria. (c) How long does it take for the population to double? 100e0.045t = 200 So, t = ln(2)/0.045 ≈ 15.4032707 The population will double in about 15.4 days. 16. Fruit flies are placed in a half-pint milk bottle with a banana (for food) and yeast plants (for food and to provide a stimulus to lay eggs). Suppose that the fruit fly population after t days is given by 230 P (t) = 1 + 56.5e−0.37t (a) Determine the initial population of the fruit flies. P0 = 230 57.5 ≈ 4 fruit flies. (b) What will the population be after 5 days? In 5 days there will be about 23 fruit flies. (c) How long does it take for the population to reach 180 fruit flies? 230 180 = 1+56.5e −0.37t 5 ln( 18·56.5 ) t = −0.37 ≈ 14.3653364 So the population of fruit flies will reach 180 in about 14.4 days. 9 10