MATH 165 Common Final Exam Review SPRING The amount of carbon 14 remaining in animal bones after t years, is given by

Transcription

1 1 MATH 165 Common Final Exam Review SPRING The amount of carbon 14 remaining in animal bones after t years, is given by A() t t A0e where 0 A is the amount present initially. Estimate, to the nearest year, the age of a bone that contains % of the carbon 14 of a comparable living sample.. Under ideal conditions a certain population of bacteria doubles in size every 4 hours. After 14 hours, there are approximately 10,000 bacteria. What was the approximate initial population of bacteria? 3. Solve the equation. Express answers in exact form x 3x 4. Solve the equation: log xlog ( x6) 4 5. Solve the equation: x 4x 6. Express as a single logarithm with positive exponents: 1 loga x loga y 3log a z 7. Expand the logarithm: log 8 4 y 3z x 3 8. Sketch the graph of the conic section with the equation 3x y 4x 6y 45 0 and state the vertices, foci, center, asymptotes, directrix, latus rectum (if appropriate). x y 9. Given 1, sketch the graph and and state the vertices, foci, center, asymptotes, 9 7 directrix, latus rectum (if appropriate). 10. Given ( y ) 4x 0, sketch the graph and and state the vertices, foci, center, asymptotes, directrix, latus rectum (if appropriate).

2 11. Given vectors u and v in the figure, find w u v and do the following: a. Express w in component form b. Express w in terms of i and j c. Find the magnitude and direction of w 1. Find the first 5 terms of the sequence defined by n 3 an ( 1). n Find the sum of the infinite series Find the sum of the infinite series The third term of an arithmetic sequence is 13 and the sixth term is 31. Find the fifth term, a Solve the right triangle. 61 o 17. A woman standing on a hill sees a flagpole that she knows is 60 ft. tall. The angle of depression to the bottom of the pole is 14, and the angle of elevation to the top of the pole is 18. Find her distance x from the pole.

3 3 18. If the point ( 1, 3) lies on the terminal side of, find the following: sin, cos, tan, csc, sec, cot 19. Find the exact value of each the following: 7 a. sin 6 7 b. cos 6 c. tan5 4 d. tan 3 e. sin cos 6 3 f. csc 3 g. 7 cos 1 0. Find the exact value of each the following: a. sin b. tan 3 c. d. cos 1 1 cot cos e. sin tan 1 f. g. coscos secsin If 1 sin and sec 0, determine the exact value of cos and csc. 4. Find the amplitude, period, any roots, and the equation for all asymptotes of the following functions, and sketch one period of the graph. a. y 3sin x b. y tan 3x 3. Prove the identity: a. b. c. sin 1 cos (1 cos 0) 1 cos sin( ) 1 cottan sincos tan cot 1 sec csc

4 4 4. Find all real number solutions for the following trigonometric equation: sin( ) 3cos 0 5. Solve the equation on the interval [0, ) : a. 8sin 10sin 3 b. 1sin x cos x 6. Solve each triangle. Round your answers to the nearest hundredth. a. b. 50 o 40 o 7. Solve for x to the nearest hundredth. 40 o 8. For the curve given by the parametric equations a. Sketch the curve b. Find a rectangular equation for the curve. 9. Given v 4 4 iand wcos isin 6 6, a. Write v in polar form. b. Write the exact value of vw in polar form. 1 x t 1, y t c. Write the exact value of v in polar form. w d. Find the exact values of the cube roots of v in polar form. e. Find the exact value of 4 w in rectangular form. 30. Write x y 4 in polar form.

5 5 31. Write r 1 cos in rectangular form. 3. Sketch the graph of each polar equation: a. r b Prove that 4 n n( n 1) for all positive integers n. Note: 4 n k. n k1 34. Suppose that an object attached to a coiled spring is at its resting position and moving up at time t = 0. Its maximum displacement from its resting position is 10 inches, and the time for one oscillation is 5 seconds. Assuming that the motion is simple harmonic, develop a model that relates the displacement d of the object from its rest position after time t (in seconds) Find the partial decomposition of the rational expression: x 5x6

6 6 ANSWERS 1. 1,618 years log log log /7 3 xz 6. log a y log y log x 3log3 6log z 8. Center: (4,-3) Vertices: 4, 3 3 Foci: 4, 3 9. Center: (0,0) Vertices: (3,0), (-3,0) Foci: (6,0), (-6,0) Asymptotes: y 3x

7 7 10. Vertices: (-5,) Foci: (-4,) Directrix: x a. w 7, b. w 7i j c. w 53 ; tan ,,,, o a ft, c ft, A 9 o 17. x ft sin 3, cos, tan csc, sec 10, cot a. sin b. cos, c. tan d. tan 3 e. sin cos 0 f. csc g. cos 1 4

8 a. sin or 30 b. tan 3 or 60 c. cos or d. cot cos e. sin tan 1, f.cos cos g. sec sin cos, csc a. Amplitude: 3, Period:, Roots: n, n any integer y o o o x n n b. Period:, Roots:, n any integer, Asymtotes:, n any integer y x

9 9 3. proof 4. k, where k is any integer 5. a. 0.57,.8889 radians b. 5 0,,, a o o C, B 30.71, c 19.3 b. a 1.17 cm, B o, C x o 9. a. x 4 b. y, x, a. 3 3 v8cos isin 4 4 b wv 16cos isin 1 1 c. v 7 7 4cos isin w 1 1 w cos isin w cos 11 isin 11 w cos 19 isin d e. w 16cos isin 3 3

10 r 4sec 31. y 4x 1 3. a. b. 33. proof 34. d t 10sin t x 5x 6 x 3 x