5. The angle of elevation of the top of a tower from a point 120maway from the. What are the x-coordinates of the maxima of this function?

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1 Exams,Math 141,Pre-Calculus, Dr. Bart 1. Let f(x) = 4x+6. Find the inverse of f algebraically. 5x 2. Suppose f(x) =x 2.We obtain g(x) fromf(x) by translating to the left by 2 translating up by 3 reecting over the x-axis and stretching vertically by a factor of 20. Write down an equation for g(x) 3. Suppose = 17, in what quadrant does the terminal side of the the angle lie? 3 Express in degrees. 4. Suppose we are given the point P (3 ;7) on the terminal side of an angle. Find sin(), cos() and tan(). 5. The angle of elevation of the top of a tower from a point 120maway from the building (on level ground) is 65 o. Find the height ofthetower. 6. Let y = 2 cos(4x) What are the x-coordinates of the maxima of this function? x = Sketch the graph over a three-period interval. Label the x- and y-axes appropriately. 7. Given f(x) = p x +3,sketch a graph of f(x) and nd f ;1 (x) graphically. 8. Find the horizontal and vertical asymptotes for the function g(x) = 3x;1 2x+2. Give a sketch of the graph, indicating the asymptotes. (You can use your calculator for this.) 9. If s =7in and r =12in, whatis? (give appropriate units) 10. Give the exact values of the following: (answers in decimal form will receive no credit!) tan( 6 )= sin( 3 )= cos( 3 4 )= 1

2 11. (10 pts) Suppose a wheel has a radius of 8 in. If the angular velocity is given by 5 rotations per minute, nd the linear velocity in feet per hour. 12. Suppose tan x = 5, nd the other ve trigonometric functions. (Assume x lies 12 in the rst quadrant.) 13. Use the calculator to solve sin x = :25 if 0 x Let f(x) =3cos(2x) ; 2sin(2x). Sketch a graph of f(x), and estimate the values of a b and h so that f(x) a sin[b(x ; h)]. 15. Graph the function y = sin ;1 (3x). Label the axes and state the domain and range of the function in symbolic notation. (no decimals.) 16. Find an algebraic expression equivalent to tan(cos ;1 ( 1 x )). 17. A tree casts a shadow 51 ft. long when the angle of elevation of the sun (i.e. the angle measured with the horizon) is 25 o 12 0.How tall is the tree? 18. Simplify the following trigonometric expresion to a single term. 19. Combine the fractions and simplify: 20. Conrm the following identity: 21. Conrm the following identity: sec 2 x ; tan 2 x cos 2 y +sin 2 y 1 1 cot 2 x ; cos 2 x tan 2 x ; sin 2 x =sin 2 x tan 2 x sin 2 x cos 3 x = (sin 2 x ; sin 4 x)(cos x) 22. Let f(x) = 4x+6 5. Find the inverse of f algebraically. 23. Indicate underneath each of the following functions if they are one-to-one or not. 2

3 24. Suppose f(x) =x 2. We obtain g(x) fromf(x) by translating to the left by 8 translating up by 7 reecting over the y-axis and stretching vertically by a factor of 2. Write down an equation for g(x). 25. Graph the inverse of the following function, and indicate the domain and the range of the inverse function. 26. Find all horizontal and vertical asymptotes for R(x) = 3x2 ;1 x 2 +5x+6. Explain. 3

4 27. Suppose = 5, in what quadrant does the terminal side of the the angle lie? 2 Express in degrees. 28. Suppose we aregiven the point P (6 ;12) on the terminal side of an angle. Find sin(), cos() and sec(). 29. Use a right triangle to determine the values of all the trigonometic functions of, where tan() = Suppose we have a circle of radius 2. Now consider an angle of 2. What is the 3 length of the arc dened by the angle and the circle? 31. The angle of elevation of the top of a building from a point 100 m away from the building (on level ground) is 78 o. Find the height of the building. 32. Give the period, domain, range, asymptotes, zeroes of the function y = sec x. Sketch the graph of the function. Do the same for y = 2 sec 3x. 33. Conrm the following identity : 34. Conrm the following identity : cos 3x = 4 cos 3 x ; 3 cos x cos 2 ( x 2 )=1+secx 2 sec x 4

5 35. Find all solutions in the interval [0 2) : sin 3x =sinx 36. Let y = 2 cos(4x) What are the x-coordinates of the maxima of this function? x = Sketch the graph over a three-period interval. Label the x- and y-axes appropriately. 37. Let y = 5 sin(3x) What are the x-coordinates of the minima of this function? x = Sketch the graph over a three-period interval. Label the x- and y-axes appropriately. 38. On the left sketch a graph of the cosine function.on the right sketch a graph of the secant function. Let y = ; sec(4x) What are the asymptotes of this function? x = What eect does the `-' sign have on the graph of the function? Sketch the graph over a two-period interval. Label the x- and y-axes appropriately. 39. On the left sketch a graph of the sine function.on the right sketch a graph of the cosecant function. Let y = 4 csc(3x) What are the asymptotes of this function? x = What eect does the `4' have on the graph of the function? Sketch the graph over a two-period interval. Label the x- and y-axes appropriately. 40. Recall that the perimeter of a rectangle is P =2W +2L, where W and L are the width and the length of the rectangle. Area is given by A = WL Suppose that a rectangle is 5 ft longer than it is wide and its perimeter is 38 ft. Find the width, the length and the area of the rectangle. 5

6 41. How much 10% solution and how much 25% solution should be used to make 100 gal of 15% solution? 42. Use your grapher to nd all local maximum and minimum values, and the value of x where they occur, for h(x) =x 2p x + 4.(Give asketch of the graph.) 43. Solve j2x ; 1j = 3 by nding x-intercepts (2 decimals accuracy). (Give a sketch of the graph you use.) 44. One truck leaves 2 hrs ahead of a second truck. If the rst truck drives at a rate of 55 mph and the second travelsat65mph,how long will it take for the second truck to catch the rst one? 45. How would one obtain y =(x ; 1) from the function f(x) =x 2? 46. Graph both h 1 (x) =;(x ; 1) 2 +6andh 2 (x) =;((x ; 1) 2 + 6). Describe how you would get h 1 (x) and h 2 (x) via translations and reections from f(x) =x Let f(x) =jx ; 2j + jx +4j. Graph this function on the standard screen of your calculator. Sketch the graphs of y = f(x), y = f(x)+1 and y = f(2x) (Do not use the calculator to nd the latter two). 48. You are given four gures, which allgiveyou the graph of some function y = f(x). On the top-left gure sketch the graph of y = f(x) + 1. On the top-right gure sketch the graph of y = f(x - 1).On the bottom-left gure sketch the graph of y = f(2x). On the bottom-right gure sketch the graph of y = f(-x). 49. Describe how the graph of y = x 2 can be transformed to the graph of the equation y =(x ; 1) The graph of y =9x 2 can be obtained from the graph of y = x 2 byavertical stretch (or shrink) and by a horizontal shrink (or stretch). State the shrink and stretch factors. 51. Is h(x) =x 2 ; 5 one-to-one? Show your work. 52. Given f(x) = p x +3,nd f ;1 (x). 53. Suppose y = x 2. G(x) is obtained from the graph of y via a vertical stretch of a factor 3, then a shift right by 4 units. Write down an equation for G(x). 54. Describe in words how the graph of y = x 2 can be transformed into the graph of the equation y =(x ; 1) If g is the reection of f(x) =x 3 ; 5x 2 ; 3x + 2 across the y-axis, nd the equation for g. 6

7 56. Given the parametric equations x =2t;3 y = t 2 ;3. Write a pair of parametric equations that dene the inverse for the given parametric equations. 57. Determine graphically whether the function f(x) =x 2 +5 is one-to-one. (Sketch graph, and say why it is or is not one-to-one.) 58. f(x) = 2x;3 x+1 Find f ;1 (x). 59. Explain why f(x) =1+ 2 x domain. is a rational function (show work) and state its 60. Find the horizontal and vertical asymptotes for the function g(x) = 3x;1 x Change 23 o 12 0 to decimal form. 62. If s =3m and r =1m, what is? (give appropriate units) 63. Express -6 in degrees. 7

8 43. (5pts) Assume that is an acute angle in a right triangle and that cot() = 11.Evaluate the remaining 5 trigonometric functions using the methods from (ie right triangles) 44. A guy wire from the top of a transmission tower forms a 75 o angle with the ground at a 55ft distance from the base of the tower. How tallisthetower? (Show work) 8