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Math 111 practice final exam: (From Fall 1999 final exam) General test instructions: Show all your work on this test paper! If you solve a problem algebraically show all your steps. If you solve a problem by graphing on your calculator, show a sketch of the graph, with the solution labeled. Where appropriate, round answers to 3 decimal places. Solve the following equations: 1. x 2 10 x 5 8 2. 3. 4 x 17 2 x 1 Solve this system of equations: 3 x y 15 x 2y 2 4. Solve this system of equations, find all points of intersection. x 2 y 2 169 x 2 8 y 104 5. .A store has $30,000 of inventory in 12-inch and 19-inch color televisions. The profit on a 12-inch set is 22 % and the profit on a 19inch set is 40 %. The profit for the entire stock is $10,500. What is the $ value of the inventory for each type of television? Solve the following inequalities: 6. 2 x 3 6 7. x 2 x 2 8. Evaluate: a. log 6 39 = b. log e 5 = Solve for x: 9. e( 6 x 1) 4 7 11. a. Determine the interest rate if $600 grew to $800 in 5 years assuming interest was compounded continuously. b. At that interest rate, when would the account balance be 1200? 12. a. Graph f ( x) log 3 x and g ( x) log 3 ( x 1) 2 (Label intercepts and asymptotes.) b. Describe how the graph of g(x) differs from that of f(x) in terms of transformations such as shifts up or down, left or right. c. What is the domain of g ( x) log 3 ( x 1) 2 ? 13. Sketch a graph of a 3rd degree polynomial with a leading coefficient that is negative. 14. a. Graph f ( x) x 4 2 x 2 10 . Clearly label all important aspects such as intercepts and maximum or minimum points. b. Over what intervals is this graph increasing? 15. Find the number of units, x, that would have to be sold to produce a maximum revenue, R, where R 01 . x2 70x 25000 . b. What is the maximum revenue? 2x 1 16. Analyze the function f(x) = x (a) y-intercept (b) x-intercept (c) vertical asymptotes (d) horizontal asymptote 10 log 4 ( x 1) log 4 ( x 2) 1 (e) graph y = f(x); include x and y-intercepts and all asymptotes. 17. Multiply these complex numbers, simplify the answer and write it in the standard a+bi form. (3 + 2 i)(6 – 5 i) 18.Find all the zeros of the polynomial function f ( x) 4 x3 20 x 2 25x 19. f ( x ) is sketched on the axes below. Translate it to sketch a graph of f ( x 2) 1 23. The approximate number of new AIDS cases reported in each of the years 1983-1986 is given in the table, with year 3 corresponding to 1983. Year(3=1983) 3 New AIDS Cases 2100 4 5 6 4400 8200 13100 a. Use your graphing calculator to fit a linear model to the data. Write the equation and correlation coefficient here. b. Use your graphing calculator to fit an exponential model to the data. Write the equation and correlation coefficient here. 20.Given the function g ( x) 2 x , what is it’s a. domain? b. range? c. inverse function? d. Graph g ( x) and g 1 ( x) c. Which equation is a better model of the data? Explain. 21. Given f ( x ) = x 2 and g ( x ) = 4 x 2 find: a. f (5) b. g (-2) c. ( g f )( x ). 22. Find an equation of the line that passes through the point (1, 2) and is perpendicular to the line 3x - 2y = 5. d. Using the better equation, estimate how many new cases of aids there were in 1987.