0 where A is the amount present initially. Estimate, to the nearest year, the

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1 MATH 65 Common Final Exam Review SPRING 04. Cindy will require $9,000 in 4 years to return to college to get an MBA degree. How much money should she ask her parents for now so that, if she invests it at 9% compounded continuously, so she will have enough for school? Round your answer to the nearest dollar. You must write the model first. Show your work to get full credit.. During 99, 00,000 people visited Rave Amusement Park. During 997, the number had grown to 84,000. If the number of visitors to the park obeys the law of uninhibited growth, find the exponential growth function that models this data. You must write the model first. Show your work to get full credit. Video for and. The amount of carbon 4 remaining in animal bones after t years, is given by 0.000t At () = Ae 0 where A is the amount present initially. Estimate, to the nearest year, the 0 age of a bone that contains % of the carbon 4 of a comparable living sample 4. Solve the equation. Express answers in exact form 5 x = x 5. Solve the equation: log (x ) log ( x 5) = 4 6. Solve the equation: log (5x+ 8) = log (5x ) 7. Solve the equation: 8 = 6 x 4x

2 8. Express as a single logarithm with positive exponents: loga x loga y+ loga z 9. Expand the logarithm: 4 y x log 8 z 0. Graph the function of the function. f x x+ ( ) = + and determine the domain, range, and horizontal asymptote. Sketch the graph of the conic section with the equation x y 4x 6y = and label the vertices, foci, center, and asymptotes/directrix/latus rectum (if appropriate).

3 x y. Given =, sketch the graph and label the vertices, foci, center, and 9 7 asymptotes/directrix/latus rectum (if appropriate).. Given ( y ) 4x 0 = +, sketch the graph and label the vertices, foci, center, and asymptotes/directrix/latus rectum (if appropriate). 4. 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

4 4 5. Write an equation of a parabola with focus (, 4) and directrix y =. 6. Write an equation of a hyperbola with foci (0, ± 6) and vertices (0, ± ). 7. A sunburst widow is constructed as shown below. At its highest, the window is 5 in. tall at its highest (from center), and 80 in. wide at the bottom. Find the height, h, of the window at 0 in. from center. Assume the ellipse is centered at the origin. 8. Find the first 5 terms of the sequence defined n by B n = ( ). n 6 9. Find the sum: ( k ) k = 0. Find the sum of the infinite series Video for 0-. Find the 0 th term and write the general term for the sequence 9,, 7. Find the sum of the infinite series The third term of an arithmetic sequence is and the sixth term is. Find the fifth term, a 5.

5 5 4. Solve the right triangles. a. b. 6 o 5. 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 4, and the angle of elevation to the top of the pole is 8. Find her distance x from the pole. 6. From a point 550 ft. from the base of the near edge of a building, an observer finds that the angle of elevation to the top of the buildings is 7, and the angle of elevation to the top of the flagpole to be 9.5. Find the height of the flagpole (assume that the flagpole is also on the near edge of the building). 7. If the point (, ) lies on the terminal side of θ, find the following: sinθ, cosθ, tanθ, cscθ, secθ, cotθ 8. Find the exact value of each the following: 7π a. sin 6 7π b. cos 6 c. tan 5π

6 6 d. 4 tan π π π e. sin cos 6 π f. csc g. 7π cos 9. Find the exact value of each the following: a. sin b. tan ( ) 0. If c. d. cos cot cos 5 ( ) e. sin tan ( ) f. 7 cos cos 9 g. 8 sec sin 7 sinθ = and cosθ > 0, find tanθ. 5. If sinθ = and secθ > 0, determine the exact value of the following: 4 a. cosθ b. cscθ. Find the amplitude, period, any roots, and the equation for all asymptotes of the following functions, and sketch one period of the graph. π a. y = sin 4x b. y = tan x

7 7. Write an equation of the function for the graphs given below (there are several possible): a. b. sin θ 4. Prove the identity: = cosθ + cosθ ( + cosθ 0) 5. Prove the identity: sin( α + β ) = + cotα tan β sinαcos β 6. Evaluate the following without using a calculator. Show work for full credit. o o o o sin(0 )cos(40 ) cos(40 )sin(0 ) tanθ + cotθ 7. Prove the identity: = secθ cscθ 8. Find all real number solutions for the following trigonometric equation: sin( θ) + cosθ = 0 9. For angle θ, find all values in degrees and radians over the interval [0, π ) that satisfy the following trigonometric equation: 8sin θ + 0sinθ =

8 8 40. Solve the following equation analytically, over the interval [0, π ) (Answers exact or to two decimal places): sin x= cos x 4. Solve each triangle. Give your answers to the nearest hundredth. a. b. 50 o 40 o c. a= 6, b= 8, A= 9 o 4. Solve for x to the nearest hundredth. 40 o 4. A projectile is shot at an angle of elevation of 0 with an initial speed of 96 feet per second from a height of 64 feet above the ground. a. Divide the initial velocity into its horizontal component, v 0x, and its vertical component, v 0 y. b. Write two parametric equations in the forms x= v0xt and y = 6t + v0yt+ h0. c. What is the maximum height reached by the projectile? d. How far does the projectile travel horizontally? π π 44. Given v= iand w= cos + isin 6 6, a. Write v in polar form. b. Write the exact value of vw in polar form. 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. 4 e. Find the exact value of w in rectangular form.

9 9 45. Write x y = 4 in polar form. 46. Write r = in rectangular form. cosθ 47. Sketch the graph of each polar equation: a. r =

10 0 b. θ = 5π Prove that n= nn ( ) for all positive integers n. Note: n= k. n k = 49. 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 0 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). 50. Find the partial decomposition of the rational expression: x 5x+ 6

11 . 0.9(4) 9, 000 = Pe $,56. 84,000 = 00,000e kt.,68 years log 5 4. log + log 5 5. x = 6 6. undefined 7. -6/7 P= 00, 000e 8. xz log a y 9. log y + log x log 6log z 0. Domain: all reals; Range: y < ANSWERS t. Center: (4,-) Vertices: ( 4, ± ) Foci: ( 4, ± )

12 . Center: (0,0) Vertices: (,0), (-,0) Foci: (6,0), (-6,0) Asymptotes: y = ± x. Vertices: (-5,) Foci: (-4,) Directrix: x = 6 4. a. w = 7, b. w= 7i j o c. w = 5 ; θ = tan = ( x+ ) = ( y+ ) y x 6. = 4 7. Approx..65 inches 8.,,,, /7

13 . A = 9 + ( n )( 6) ; 95 n. S =. a 5 = 5 4. a. a= 7.4 cm, A= 54.4 o, B= 5.6 b. a = 6.9 ft, A= 9 o, c= 8. o o ft 6. x = 8.9 ft 7. 0 a. sinθ = b. cosθ = 0 0 d. cscθ = 0 0 e. secθ = 0 f. 8. 7π a. sin = 7π b. cos 6 = 6 4π d. tan = e. π π sin cos = 0 f. 6 7π 6 g. cos = 4 c. tanθ = cotθ = c. tan 5π = 0 π csc = 9. π a. sin = 6 6 d. cot cos = g. sec sin = 7 6 b. tan ( ) ( ) e. sin tan ( ) π = c. = f. π cos = 7 7 cos cos = a. cosθ = b. cscθ = 4 5 5

14 4.. tanθ = 4 a. Period: Amplitude: n + Roots: x=, for n= any integer b. π Period: Amplitude: none nπ Roots:, for n = any integer Asymptotes: π nπ x= +, for n= any integer 6. a. f( x) =.5sin( π x) b. f( x) = sin x+ sin θ cos θ ( + cos θ)( cos θ) 4. = = = ( cos θ) = + cosθ = cosθ + cosθ + cosθ + cosθ sinαcos β + sin βcosα sinαcos β sin βcosα sin βcosα = + = + = + cotα tan β sinαcos β sinαcos β cos βsinα cos βsinα o o o o o o o sin(0 ) cos(40 ) cos(40 ) sin(0 ) = sin(0 40 ) = sin( 0 ) =

15 5 sinθ cosθ sin θ + cos θ + tanθ + cotθ sin cos cos sin 7. cosθ sinθ cosθsinθ θ + θ θ θ = = = secθcscθ = cosθsinθ cosθ sinθ cosθsinθ sin( θ) + cosθ = 0 sinθcosθ + cosθ = 0 8. cos θ(sinθ + ) = 0 cosθ = 0 OR sinθ = π π θ =, never π θ = + πk, where k is any integer 9. 8sin θ + 0sinθ = 8sin θ + 0sinθ = 0 (4sinθ )(sinθ + ) = 0 sinθ = OR sinθ = 4 θ = sin never 4 θ = 0.57,.8889 radians o o θ = ,65.58 π 5π 40. 0,,, π a o o C =, B= 0.7, c= 9. b. a=.7 cm, B= 08.4 o, C =.86 o o c. B= 7., C =.7, c =.0 4. Approx o 4. a. v 0x v 0y b. = 96cos0 and = 96sin 0 = 48 x= t y = t + t+ (96cos0 ) and

16 6 c. 00 feet d..55 feet x r r r a. π π v= 8 cos + isin 4 4 b. π π wv = 6 cos + isin c. v 7π 7π = 4 cos + i sin w w cos π isin π w cos π isin π w cos 9π isin 9π = + = + = d. 0 4 π π e. w = 6 cos + isin y = 4 cos θ r sin θ = 4 cos θ = 4 = 4sec θ r = cosθ r rcosθ = x y x + = 47. a. b.

17 7 () n= nn ( + ) 48. n ( ) ( i ) = nn ( + ) i= Using first formula () Step. Prove formula for smallest number. Substitute n = to both sides n= nn ( + ) So () = ( ()+) = Step. Assume formula true for n = k, i.e., k= kk ( + ) Step. Prove formula true for n = k+, i.e., k+ ( k+ ) = ( k+ )( k+ ) Starting with left side associate the first k terms, i.e., ( k) + ( k+ ) = kk ( + ) + ( k+ ) by induction hypothesis, but kk ( + ) + ( k+ ) = ( k+ )( k+ ) by factoring [end of proof] Using second formula () Step. Prove formula for smallest number. Substitute n = to both n sides ( i) = nn ( + ) So = ( i) i= = )()(() + ) = i= Step. Assume formula true for n = k, i.e., ( i) k i= k + Step. Prove formula true for n = k+, i.e., ( ) i= = kk ( + ) i = ( k+ )( k+ ) Starting with left side associate the first k terms, i.e., ( ) So by induction on first term we have k+ k k+ i = i+ i i= i= i= k+

18 8 k k+ i+ i= k(k+) + i= i= k+ k + i= k+ i = k(k+) + (k+) = (k+)(k+) [end of proof] 49. π st ( ) = 0sin t = x 5x+ 6 x x