MAT 111 Final Exam A Fall 2015 Name:

Transcription

1 MAT 111 Final Exam A Fall 2015 Name: Show all work on test to receive credit. Draw a box around your answer. If solving algebraically, show all steps. If solving graphically, sketch a graph and label the solutions. Where appropriate, round correctly to 2 decimal places. Be sure to check your solutions. 200 possible points

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3 Fall 2015 MAT 111 Final Exam Name: Score: /200 Show all work on test to receive credit. Draw a box around your answer. If solving algebraically, show all steps. If solving graphically, sketch a graph and label the solution(s). Where appropriate, round correctly to 2 decimal places. 1. [6 pt.] Solve the equation for x. 4x 9 = 17 3(2x 1) 3. [6 pt.] Solve the equation for x x x = 4 2. [6 pt.] Solve for x. Find the exact value. 3e 6x+7 2 = [6 pt.] Solve the system of equations. 5x y = 21 { 2x + 3y = 12

4 5. [8 pt.] Solve the following inequality. Write the solution in interval notation. 11 4x > [6 pt.] Solve the equation for x. log 4 x + log 4 (x 6) = 2 6. [8 pt.] Solve the following inequality. Show your work by using the number line below. Write the solution in interval notation. (x 3) (x 2)(x+1) 0 8. [6 pt.] Solve the equation for x. ( 1 5 )3x 2 = 125

5 9. [8 pt.] Find all real and complex solutions and simplify completely. x 2 = 8x Sketch the graphs of the following. Label asymptote(s) and at least 2 key points on each graph. a. [6 pt.] y = ln (x) y = ln (x + 2) b. [6 pt.] 10. [6 pt.] Write the following as a sum and difference of simple logs. Simplify completely and write powers as factors. y = 2 x x + 8 log 4 [ 64(x 2) 3] y = 2 x

6 12. Given the function f(x) = 2x+1 x Given f(x) = 6 x 2 3 and g(x) = x + 3 a. [6 pt.] Find the inverse function f 1 (x). a. [3 pt.] Find f(g(x)). b. [3 pt.] Find g(f( 2)). b. [2 pt.] What is the range of f 1 (x)?

7 14. Given f(x) = x 3 + 2x 2 + 6x 5 a. [4 pt.] Use a graphing calculator to sketch the graph of f(x). 15. The graph of the linear equation f(x) is given below. f(x) ( 3,2) (0,4) b. [2 pt.] State the local minimum point(s). a. [2 pt.] Find the slope of f(x). c. [2 pt.] State the local maximum point(s). b. [4 pt.] Find the equation of the line that is parallel to f(x) and passes through the point (4, 3). Write your answer in slope-intercept form. d. [2 pt.] On what interval(s) is f(x) increasing?

8 16. Erik the alien Viking mathematician has built a catapult to launch rocks into the sea off a cliff. He has determined that his height h of a rock t seconds after launch is given by the equation 17. Consider f(x) = 5x 3 (x 4)(x + 3) 2 a. [2 pt.] The end behavior of the graph of f(x) resembles the graph of what power function? h(t) = 2t t where h is in meters (above alien sea level) and t is in seconds. Without using your calculator to graph the function answer the following a. [4 pt.] What is the maximum height reached by the rock? b. [9 pt.] Find the zeros of f(x). Determine the multiplicity of each zero. State if the graph touches or crosses the x-axis at each zero. Zeros Multiplicity Touch/Cross b. [4 pt.] How long after launch does the rock pass a height of 177 meters on its descent? c. [5 pt.] Sketch the graph of f(x). Label all zeros.

9 18. Consider the function f(x) = x2 3x 10 x 2 = (factored form) [2 pt.] 5x a. [2 pt.] Find the domain of f(x). b. [1 pt.] Find the x-intercept(s) of f(x). c. [1 pt.] Find the y-intercept of f(x). d. [2 pt.] Find the vertical asymptote(s) of f(x). e. [2 pt.] Find the horizontal asymptote of f(x). f. [2 pt.] Find the coordinate points for any hole(s) of f(x). g. [4 pt.] Sketch the graph of f(x). Label all intercepts, asymptotes, and holes.

10 19. [8 pt.] The manager of a store that specializes in selling tea decides to experiment with a new blend. She will mix Wrightsville s Finest that sells for $5.70 per pound with Leland s Pride that sells for $3.20 per pound to get 100 pounds of the new blend. The selling price of the new blend is to be $4.12 per pound with no difference in revenue. How many pounds of each type of tea should be used to obtain this blend? 20. The following data represents the price, p, and quantity demanded, q, per day of 24 LCD monitors. Price, p, in dollars Quantity Demanded, q We want to find the model of best fit for the data that predicts the quantity demanded of a LCD monitor as a function of the price. a. [2 pt.] Using you graphing calculator, decide whether a linear or exponential model is more appropriate. Give a reason for your answer. b. [2 pt.] Write the equation of the model of best fit. c. [2 pt.] Using the answer to part (b), estimate the quantity demanded if the price is set at $160. Round to the nearest whole number.

11 21. Bob just inherited $300,000 from his grandfather. He is going to invest half of it in a savings account at 5%, which is compounded monthly. [6 pt.] How much will Bob have saved for retirement after 28 years? 22. A sample of radioactive Stevium decays exponentially according to the following equation, where A is the amount (in grams) after t days. A = 250e 0.056t a. [3 pt.] How heavy was the initial sample? b. [3 pt.] How heavy is the sample after 7 days? c. [4 pt.] What is the half-life of Stevium? Correct to two decimal places.

12 x < x Let f(x) = { x < x 5 a. [4 pt.] Sketch f (x) labeling key points. b. [2 pt.] Find the domain of f (x). Use interval notation. c. [4 pt.] Find the range of f (x). Use interval notation. d. [1 pt.] f(2) = e. [1 pt.] f( 6) = 24. Starting with the graph of y = f(x) (pictured) a. [6 pt.] List the transformations necessary to obtain the new function. y = 1 2 f(x) 3 b. [4 pt.] Graph the new function (on the same axes). Label two points on the new function.

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