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Mathematics 142 final exam Friday, June 19, 2009 please show your work to get full credit Evaluate each of the following: North Seattle Community College ✓ ◆ ⇡ (a) sec 5 Winter Quarter 2012 (c) 60e .02x when x = 40 PRECALCULUS Practice Final 1 The terminal side ofpa central angle A in standard position intercepts the unit circle at p STUDENT NAME: __________________________ e point (x, y) = ( 1/ 5 , 2/ 5). Find the exact values (some involving square roots) of e six trigonometric functions name: of A. Mathematics 142 final exam QUIZ SCORE: ______________________________ Friday, June 19, 2009 4. your Solve fortox:get full credit please show work Solve for t: ✓ ◆ x .05x 10 log = 43 80e = 20 10 5 1. Evaluate each of the following: 1. Evaluate each of the following: If a sequence is defined by ak✓=⇡ ◆ 6(1.02)k for k 1 (a) sec (b) log3 19 (c) 60e .02x when x = 40 5 ) Write down the first four terms of (b) What kind of sequence is this? e sequence. 2. The terminal side of a central angle A in standard position intercepts the unit 2. The terminal side ofpa central angle A in standard position intercepts the unit circle at circle at the point (x, p y) = (-1/ 5, 2/ 5 ). Find the values of all six trig functions. 40 the point (x,X y) = ( 1/ 5 , 2/ 5). Find the exact values (some involving square roots) of ) Find the of ak thevalue six trigonometric functions of A. 3. k=1 (a) Solve for t: 80e-0.5t = 20 (b) Solve for x: 10log (𝑥 ) = 43 10!! 4. Solve for x: Solveoffor t: indicated angle, side length, or arclength: Find the3.value each ✓ ◆ st 4. If a sequence is defined by ak = 6 (1.02)k for k≥1, then xwrite=the .05x 10 log 431 six terms of 80e = 20 insert diagrams here 5 the sequence. 10 (b) log3 19 k 5. Findisalldefined the solutions θ between radians for each of the following If a sequence by2⇡akradians =of 6(1.02) for k0ofand 1the2πfollowing Find all5. solutions between 0 and for each trigonometric trigonometric equations: uations: (a) Write(a) 2 θfirst down the 6 cos = 3 four terms of (b) (b) 3 sinWhat θ + 4kind cos θof=sequence 0 is this? the sequence. page by two ) 3 cos2 ✓ = 6 (b) 4 cos ✓ + 3 sin ✓ = 0 ( hint: divide both sides of equation cos ✓ ) 6. Expand and simplify: (a) (b) 40 X ! (c) Find the value ✓ of ak◆ e2x y 3 2⇡ Expand and simplify: 6 cos ✓ k=1 9. Expand and simplify: ln xpw 3 continues on angle, next page 6. Find the value of each indicated side length, or arclength: ! 1 3 period and horizontal shift. 7. For y = 60sin 10. (40xFor + !the ) find the insert diagrams angle ✓ amplitude, = coshere 7 8. A sum of $600 is deposited into a bank with 3% rate per annum. Find thetriangle to find the exac (b) Use your (a) Draw a right triangle illustrating 7. Find amount all solutions 0 and after 2⇡ radians forcompounding each of the following trigonometric in thebetween bank account 10 years value of ✓ ◆ the angle ✓. (Two (b) side lengths of the trianequations:(a) monthly, 1 3 continuously. tan cos gle should be whole numbers) 7 (a) 3 cos2 9. ✓ =Show 6 that ln(sec θ(b) 4 cos ✓ + 3 sin ✓ = 0 ( hint: divide both sides of equation by cos ✓ ) (your answer will involve a square root) + tan θ) + ln(sec θ – tan θ) = 0 ✓ ◆ ✓ ◆ 2⇡ ⇡ 8. Expand and simplify: 611. cos For ✓ the sinusoidal function y = 60 sin 40x + 3 6 continues on next page (a) Compute the amplitude, period, and horizontal shift. (b) Use your calculations to draw a graph of the sinusoidal function, showing the location of quarter-cycle points. 10. Describe for which values of x or θ each equation is valid (a) cos(cos-1 x) = x (b) cos-1 (cos θ) = θ 33) The roof of a building is in the shape of the hyperbola y2 - x2 = 50, where x and y are in meters. Refer to the figure and determine height h of theyoutside 11. Solve for y, andthe simplify: log = 3x –walls. log N 12. Write down equations for any asymptotes each graph has. It is not necessary to actually sketch the graph. (a) f(x) = log(9-7x) a13. = bFind = 6 mthe A) 9.3 m (b) g(x) = tan(x/100) equation for the following in their standard forms: C) -14 m B) 86 m D) 44 m (a) Ellipse: length of the minor axis is 16 and the foci are located at 34) The of a 4). building is in the shape of the hyperbola y2 - x2 = 18, where x and y are in meters. (1±roof 105, Determine the distance, w, the outside walls are apart, if the height of each wall is 11 m. (b) Hyperbola: centered at (0, 0) with vertices at (0, ±3) and a focus at (0, 34). (c) Parabola: Opening to the left with the vertex at (−6, 3) and passing through (−14,−9). !! !! 14.A)A20.3 comet follows the hyperbolic path described by − !"= 1, where x m B) 103 m C) 10.15 m!" D) 11.8 m and y are in millions of kilometers. If the sun is the focus of the path, how x2 y2 close the sun is the vertex of the path? 35) A cometto follows the hyperbolic path described by = 1, where x and y are in millions. If the sun is the 24 13 focus of the path, how close to the sun is the vertex of the path? A) 1.2 million B) 6.1 million C) 4.9 million D) 37 million 15. For the arithmetic sequence 100,80,60,40,..., find an expression for an, calculate a80 , and calculate S80 . 16. Find patterns and identify the nth term for each of these patterns: (a) 3, 15, 27, 39, 51, ... (b) 4, -16, 64, -256, 1024, ... 17. Find the sum of the sequences: ! 𝑎 6 !!! ! 𝑏 𝑘 !!! ! 𝑐 4𝑘 − 5 !!! ! 𝑑 𝑘(𝑘 + 10) !!! 18. A new piece of equipment cost a company $53,000. Each year, for tax purposes, the company depreciates the value by 25%.What value should the company give the equipment after 6 years? 19. A pendulum bob swings through an arc 70 inches long on its first swing. Each swing thereafter, it swings only 85% as far as on the previous swing. What is the length of the arc after 12 swings? Round your answer to two decimal places, if necessary. 20. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. ! ! (a) 5 + ! + !" + ⋯ ! (b) !!! 3 4 2 !!!