Math 1324 – Final Exam Sample. 1. The revenue generated by selling x units of a product is given by the quadratic function Rx 0.03x 2 63x , where the revenue R ( x ) is measured in dollars. How many units to be sold to obtain maximum revenue, and what is this maximum revenue? Answer. This is a quadratic function concave downward. The maximum value is b 63 1,050 and indicated by its vertex (h , k ) where h 2a 2 0.03 k R1,050 33,075 . R( x) 0.03x There are 1,050 units to be sold to obtain the maximum revenue of $33,075. 2. Let the revenue function be Rx 0.03x 2 63x and the cost function be C x 12 x 1,500 , where x represent the number of units of a product being produced and sold. Find the average cost function and the profit function. Answer. Total cost 12 x 1,500 number of units x 2 Profit Revenue - Cost 0.03x 63x 12 x 1,500 0.03x 2 51x 1,500 Average cost 3. Given f x 2 x 3 and g x x 3 1 . Find f g . Answer. f g x f g x f x 3 1 2x 3 1 3 2x 3 5 OR f g x f g x 2g x 3 2x 3 1 3 2x 3 5 4. Given a. b. c. Px 2 x 2 x 2x 3 . Find the x-intercepts and y-intercept. Find the degree and the leading coefficient Sketch its graph Answer. a. x – intercepts : ( 0 , 0 ) , ( 2 , 0 ) , and (– 3 , 0 ). b. Degree is 4. Leading coefficient is – 2. c. 5. y-intercept: ( 0 , 0 ) Find the intercepts and the asymptotes, and then sketch a graph of x2 x2 R x 2 x 3x 1 x 2 x 3 Answer. 2 x – intercepts : ( – 2 , 0 ) . y-intercept: 0, 3 H.A. are x 3 and x 1 V.A. is y 0 6. Sketch a graph of f x 2 x1 3 . State the basic function and the transformations applied. Answer. Basic function is y 2 x . Transformations: left by 1 units, then down by 3 units 7. Sketch a graph of f x log 2 x 3 2 . State the basic function and the transformations applied. Answer. Basic function is y log 2 x . Transformations: right by 3 units, then up by 2 units Find the domain of f x log 2 5x 3 2 . 8. Answer. 5x 3 0 5x 3 3 x 5 3 D , 5 Find the inverse of f x 2 x1 3 . 9. Answer. y 2 x 1 3 y 3 2 x 1 log 2 y 3 x 1 x log 2 y 3 1 f 1 10. x log 2 x 3 1 Rewrite ln 2 2 ln x ln 5 3 ln y into a single logarithmic expression. Answer. 2x 2 2x 2 ln 2 2 ln x ln 5 3 ln y ln 2 ln x 2 ln 5 ln y 3 ln 3 ln 3 5y 5y 11. Rewrite log 7a 4 into a sum/difference of simple logarithms containing only 3b 5 single variables. Answer. 7a 4 log 5 log 7 log a 4 log 3 log b 5 log 7 4 log a log 3 5 log b 3b 12. The growth rate function of an investment is G(t ) 5 log 3 (2t 7) 3 (%) where t is the number of years. Find the growth rate after 8 years (using a calculator and change of base formula) Answer. G10 5 log 3 2 8 7 3 5 log 3 23 3 5 13. ln 23 3 17.27% ln 3 $40,000 is invested at the annual rate of 1.2% compounded quarterly. a. Find the value of the investment after 8 years. b. How long will it take so that the value is $50,000? r At P1 n nt Answer. P 40,000 r 0.012 0.012 a. A8 40,0001 4 n4 4 8 44,024.04 b. Solve for t where At 50,000 0.012 50,000 40,0001 4 4t 50,000 0.012 1 40,000 4 5 4t 1.003 4 5 4t ln ln 1.003 4 ln 5 ln 4 4t ln 1.003 ln 5 ln 4 t 18.62 years 4 ln 1.003 4t 14. Solve the following equations 10 a. 3 x 3 5e 2 b. 4 ln x 1 5 13 Answer. a. 10 3 5e x 3 2 10 15e x 3 6 16 15e x 3 16 e x 3 15 16 ln ln e x 3 15 ln 16 ln 15 x 3 x ln 16 ln 15 3 b. 4 ln x 1 5 13 4 ln x 1 8 ln x 1 2 x 1 e2 x e2 1 15. Use Gauss Elimination with Back-Substitution to solve the system below x 3z 2 3x y 2z 2x 2 y z 5 4 Answer. 1 0 3 2 1 0 3 2 1 0 3 2 3 1 2 3 R1 R 2 2 R1 R 3 5 0 1 7 11 0 1 7 11 2 2 2 2 0 2 1 4 1 4 7 8 1 0 3 2 1 0 3 2 2 R 2 R 3 R 3 /( 7 ) 0 1 7 11 0 1 7 11 0 0 7 14 0 0 1 2 x 3z 2 y 7z 11 z 2 y 7(2) 11 y 3 x 3(2) 2 x4 Solution is 4,3,2 16. Set up a linear system for the following problem A sum of $200,000 were invested in three accounts that paid simple interest at the rates of 2%, 3% and 1.5%. After one year the total interest earned was $3,256. The interest earned from the 2% account was $320 more than the interest earned from the 1.5% account. How much was invested in each account? Answer. Let x , y , and z be the amount invested in the 2% , 3% , and 1.5% respectively. Then z 200,000 0.02 x 0.03 y 0.015 z 3,256 x 0.02 x y 320 0.015 z 17. Find the optimal values of z 4 x 5 y with constraints x 0 y 0 x 3 y 15 4 x y 16 Answer. Vertices (0,5) (3,4) (0,0) (4,0) z-values 25 32 0 16 Maximum value of z is 32. Minimum value of z is 0. 18. Set up a linear programming problem for the following For a TV factory that produces LCD TV and Plasma TV we have the following data Profit ($) One LCD One Plasma 100 160 Assembling (hours) 2 5 Testing (hours) 1 2 Packaging (hours) 1 1 Each month the factory can assign at most 3000 hours for assembling, 1400 hours for testing, and 900 hours for packaging. Find the monthly optimal level of production (the number of TV of each type should be produced) Answer. Let P be the monthly profit, x and y be the number of LCD and Plasma TV produced in that month. Then Maximize: P = 100 x + 160 y (Profit) Constraints: x0 y0 2 x 5 y 3000 (Assembing) x 2 y 1400 (Testing) x y 900 (Packaging )
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