1. (10 pts.) Find and simplify the difference quotient, h 0for the given function

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1 MATH 1113/ FALL 016 FINAL EXAM Section: Grade: Name: Instructor: f ( x h) f ( x) 1. (10 pts.) Find and simplify the difference quotient, h 0for the given function h f ( x) x 5. (10 pts.) The graph of the function f (x) is shown below. On the same xy- plane, use transformation rules to graph g ( x) f ( x 3) 1. Label three points on the graph of the transformed function.

2 3. (10 pts.) Find the inverse function of x 7 f ( x). x 4. Given P(x) = (x + 3) (x + ) 7 a. (4 pts.) What is the degree of the function? b. (6 pts.) Determine the zeros of the function. State the multiplicity of each zero and if the graph crosses the x-axis or touches the x-axis and turns around, at each zero. Zero Multiplicity Graph crosses the x-axis, or touches the x-axis and turns around a. Crosses the x-axis b. Touches the x-axis and turns around a. Crosses the x-axis b. Touches the x-axis and turns around t 5. The function A ( t) 8.3e, models the population of Germany where A (t) is the population in millions, t years after 01. a. (4 pts.) What was the population of Germany in 01? b. (6 pts.) In which year will the population of Germany be 9.1 million?

3 6. (8 pts.) Use the properties of logarithms to expand the logarithmic expression as much as possible. 3 x y ln ( z 1) 4 7. Simplify each expression without using a calculator. a. (5 pts.) ln ( x) e b. (5 pts.) log 7 7 e 8. (10 pts.) Solve the logarithmic equation log (1x + ) log (x) = 3 9. (10 pts.) Suppose the reference angle for an angle is 60, and the terminal side of lies in quadrant IV. Find the exact value of sin( ). 10. Answer the following: 16 a. (5 pts.) Draw the angle = in standard position. 3 b. (5 pts.) State an angle between 0 and which is coterminal to the angle.

4 x 11. Given the rational function r ( x), ( x 4) ( x 1) a. ( pts.) Find the vertical asymptotes of the graph of r (x), if any. b. ( pts.) Find the x intercepts of the graph of r (x). c. ( pt.) Find the y intercept of the graph of r (x). d. ( pt.) Find the horizontal asymptote of the graph of r (x), if there is one. e. (4 pts.) Use parts (a d) to sketch the graph of r (x). Label the vertical and horizontal asymptotes Find the exact values of the following. 17 a. (5 pts.) tan 3 3 b. (5 pts.) sin 4

5 13. The Leaning Tower of Pisa is m tall. The top edge of the tower is 5.45m out from the bottom edge. Rounded to the nearest whole number, find the measure of the angle created between the ground and the tower? (Note that this is the angle in the following diagram.) 5.45 m 58.36m - f ( x) 3sin 3x 14. Given the trigonometric function a. ( pts.) Determine the amplitude of f(x). b. ( pts.) Determine the period of f(x). c. ( pts.) Determine the phase shift of f(x) d. (4 pts.) Sketch one period of the graph of f(x).

6 15. Find the exact value of the following expressions. 1 a. (4 pts.) tan b. (4 pts.) sin 1 sin Use an addition, subtraction or half angle formula to find the exact value of sin (15 ) If sin( t ), and t is an angle in quadrant III, find the exact value of the following. Show your work. 4 a. (6 pts.) cos( t ) b. (4 pts.) cos t

7 (sin x cos x) (cos x sin x) 18. (10 pts.) Verify the following trigonometric identity. sec x csc x sin x cos x (10 pts.) Find all solutions to the following equation on the interval [ 0, ). Express your answers exactly. sin x sin x 0-0. Refer to the figure below, depicting a triangle with sides and angle shown. a. (6 pts.) Use the Law of cosines to determine the length of the side c to the nearest tenth. b. (4 pts.) Use the law of sines to determine the measure of the angle A to the nearest tenth of a degree.

8 Addition and Subtraction Formulas: sin( u v) sin ucosv cosusin v sin( u v) sin ucosv cosusin v cos( u v) cosucosv sin usin v cos( u v) cosucosv sin usin v tanu tanv tan( u v) 1 tanu tanv tanu tanv tan( u v) 1 tanu tanv Double Angle Formulas: sin( u) sin ucosu cos( u) cos u sin u cos(u ) cos u 1 cos( u) 1 sin u tanu tan( u) 1 tan u Half-Angle Formulas: u sin u cos Law of Sines: 1 cosu, (± depends on Quadrant) 1 cosu, (± depends on Quadrant) u 1 cosu sin u tan sin u 1 cosu a sin A b sin B c sin C, where a, b, c are lengths of sides and A, B, C are the opposite angles. Law of Cosines: a b c bc cos A, where a, b, c are lengths of sides and A is the angle opposite side a. b a c accosb, where a, b, c are lengths of sides and B is the angle opposite side b. c a b abcos C, where a, b, c are lengths of sides and C the angle opposite side c. Arc Length formulas: s r, where is a central angle.