Math 180 - Final Review Name Find the center (h, k) and radius r of the circle. Graph the circle. 1) Find the center, radius, and graph of x + y - x + 8y + 11 = 0. 1) Use a graphing utility to approximate the real solutions, if any, of the equation rounded to two decimal places. ) x - 3x + x + 15 = 0 ) Solve the equation. 3) 1-9 5x = 7 3 3) ) x3-8x = 1x ) 5) 3n = -1n - 5 5) 6) p - 5p + = p - 3 6) 7) (x - 5) 1/ = 11 7) 8) Find the value of f(x + h) - f(x), h 0 for the function f(x) = x + 5x. 8) h Give the domain of the function. x + 7 9) f(x) = (x + 5)(x + 7) 9) 10) f(x) = 1 - x 10) Find the average rate of change for the function over the given interval. 11) f(x) = x + 1x between x = 6 and x = 8 11) Determine if the given function is even, odd, or neither. 1) f(x) = x3 - x 1) Find the requested function. 13) Let f(x) = 16 - x, g(x) = - x. Find (f + g)(x). 13) Find the indicated composite for the pair of functions. 1) (g f)(x): f(x) = x + 5x + 3, g(x) = 5x - 5 1) 15) An open box with a square base is required to have a volume of 7 cubic feet. Express the amount A of material used to make such a box as a function of the length x of a side of the base. 15) Find the vertex and axis of symmetry of the graph of the function. 16) f(x) = x + x - 1 16) 1
17) The owner of a video store has determined that the cost C, in dollars, of operating the store is approximately given by C = x - 3x + 750, where x is the number of videos rented daily. Find the lowest cost to the nearest dollar. Answer the question. 18) For the polynomial f(x) = (x - )3(x - 3)(x - ) (a) Find the x- and y-intercepts of the graph of f. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) End behavior: find the power function that the graph of f resembles for large values of x. (d) Use a graphing utility to graph the function. Determine the number of turning points on the graph. (e) Approximate the local minima rounded to two decimal places. (f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f. 17) 18) 19) Which of the following polynomial functions might have the graph shown in the illustration below? 19) A) f(x) = x(x - )(x - 1) B) f(x) = x(x - )(x - 1) C) f(x) = x(x - )(x - 1) D) f(x) = x(x - )(x - 1) Construct a polynomial with the given properties. 0) The graph of the polynomial crosses the x-axis at - and 3, touches the x-axis at 5, crosses the y-axis at -5 and is below the x-axis between - and 3. Give the equation of the specified asymptote(s). x - 8 1) Vertical asymptote(s): f(x) = x - 16 0) 1) ) Horizontal asymptote: h(x) = 8x - 9x - 7 5x - 8x + 9 ) 3) Oblique asymptote: f(x) = x + x - x - 8 3)
Construct a rational expression with the given characteristics. ) The graph of R(x) crosses the x-axis at -1, touches the x-axis at -, has vertical asymptotes at x = - and x = 3, and has one horizontal asymptote at y = -. 5) A rare species of insect was discovered in the rain forest of Costa Rica. Environmentalists transplant the insect into a protected area. The population of the insect t months after being transplanted is 5(1 + 0.6t) P(t) = (3 + 0.0t). (a) What was the population when t = 0? (b) What will the population be after 10 years? (c) What is the largest value the population could reach? ) 5) Solve the inequality. 6) (b + 6)(b + 1)(b - 1) < 0 6) 7) (x - 1)(3 - x) (x - ) 0 7) -5 - -3 - -1 0 1 3 5 Find all real zeros of the given polynomial function P, and use the real zeros to factor P. 8) P(x) = x3 + 3x - 9x - 10 8) Find the real solutions of the equation. 9) x3 - x - 0x + 10 = 0 9) Information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. 30) Degree 6; zeros: -, + i, -3 - i, 0 30) Form a polynomial f(x) with real coefficients having the given degree and zeros. 31) Find a third-degree polynomial function with real coefficients and with zeros - and 3 + 31) i. Find the complex zeros of the polynomial function. 3) f(x) = x - 8x3 + 16x + 8x - 17. 3) The function f is one-to-one. Find its inverse. 33) Determine the equation for the inverse function of y = (x + )3-8. 33) Solve the equation. 3) (x - 3)= 6 3) Convert to logarithmic form. 35) ex = 15 35) 3
Find the value of the expression. 1 36) log 9 81 36) Find the domain of the function. 37) f(x) = log (x - 9) 37) 38) log x = 3 38) 5 39) The formula D = 6e-0.0h can be used to find the number of milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. When the number of milligrams reaches, the drug is to be given again. What is the time between injections? 39) Solve the equation. 0) log x = log + log (x - 1) 0) Solve the given logarithmic equation. 1) log(3x - ) - log(x - 5) = 1) ) During 1991, 00,000 people visited Rave Amusement Park. During 1997, the number had grown to 83,000. If the number of visitors to the park obeys the law of uninhibited growth, find the exponential growth function that models this data. 3) A fossilized leaf contains 1% of its normal amount of carbon 1. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon 1. ) 3) ) A thermometer reading 79eF is placed inside a cold storage room with a constant temperature of 35eF. If the thermometer reads 7eF in 13 minutes, how long before it reaches 57eF? Assume the cooling follows Newton's Law of Cooling: U = T + (Uo - T)ekt. (Round your answer to the nearest whole minute.) 5) The logistic growth model P(t) = 180 represents the population of a species 1 + e-0.188t introduced into a new territory after t years. When will the population be 80? ) 5) In the problem, t is a real number and P=(x,y) is the point on the unit circle that corresponds to t. Find the exact value of the given trigonometric function. 33 6) - 7, ; find cos t 6) 7 7) Find the exact value of tan 7p 5p + tan. Do not use a calculator. 7) Find the exact value of the expression. Do not use a calculator. 8) cot 930e 8)
9) Find the exact value of sin 135e - sin 70e. Do not use a calculator. 9) Use the fact that the trigonometric functions are periodic to find the exact value of the expression. 50) cos 10p 3 50) 51) cos 0p 3 51) Use the given values of the sine and cosine to find the function value. 7 5) sin q = -, cos q = 3. Find cot q. 5) Find the exact value of the expression. 53) sec 65e - tan 65e 53) Find the exact value of the requested trigonometric function of q. 5) csc q = - 7 Find cot q. and q in quadrant III 5) Use the even-odd properties to find the exact value of the expression. 55) tan - p 3 55) Give the amplitude or period as requested. 56) Period of y = sin 5x 56) Write the equation of a sine function with the given characteristics. 57) Amplitude: Period: 3 57) Solve. 58) For what numbers x, 0 x p, does cos x = 0? 58) 59) For what numbers x, -p x p, does the graph of y = sec x have vertical asymptotes.? 59) Find the phase shift of the function. 60) y = sin ( 1 x - p ) 60) Graph the function. Show at least one period. 61) y = cos(3x + p ) 61) 5
Write the equation of a sine function with the given characteristics. 6) Amplitude: Period: p 6) Phase Shift: 3 Find the value of the expression. 63) sin-1-0.5 63) 6) cos-1-6) Find the exact value of the expression. 65) cos-1 cos 7p 6 65) 66) cos sin-1 5 66) 67) sec-1 () 67) Find the exact value by using a sum or difference identity. 68) sin p 1 68) Use trigonometric identities to find the exact value. 69) cos 5p 1 cos p 5p + sin 1 sin p 69) Find the exact value of the expression. 1 - tan 80e tan 70e 70) tan 80e + tan 70e 70) Find the exact value of the expression under the given conditions. 71) sin a = - 5, 3p < a < p; tan b = - 7, p < b < p Find cos (a + b) 71) Complete the identity. 7) sin (a + b) cos b - cos (a + b) sin b 7) Find the exact value of the expression. 73) cos tan-1 3 - sin -1 3 5 73) Find the exact value of the expression under the given conditions. 7) sec q = - 5 1 1, csc q > 0 Find sin (q). 7) 6
Find the exact value by using a half-angle identity. 75) tan 75e 75) Solve the equation for solutions in the interval 0 q < p. 76) cos q = 1 76) 77) sinx - cosx = 0 77) 78) 6 cos q - 7 cos q - 3 = 0 78) 79) cos x = - cos x 79) Find the missing parts of the triangle. 80) a = e30' b = e10' a = 16.30 81) b = 11.6e b = 5.69 a = 9.3 80) 81) 8) g = 15.1e a = 6.90 b = 11. 83) Two surveyors 180 meters apart on the same side of a river measure their respective angles to a point on the other side of the river and obtain 5e and 68e. How far from the point (line-of-sight distance) is each surveyor? Round your answer to the nearest 0.1 meter. 8) A straight trail with a uniform inclination of 15e leads from a lodge at an elevation of 1000 feet to a mountain lake at an elevation of 7800 feet. What is the length of the trail (to the nearest foot)? 8) 83) 8) 85) In flying the 8 miles from Champaign to Peoria, a student pilot sets a heading that is 1e off course and maintains an average speed of 136 miles per hour. After 15 minutes, the instructor notices the course error and tells the student to correct his heading. Through what angle will the plane move to correct the heading and how many miles away is Peoria when the plane turns? 85) 86) The distance from home plate to dead center field in Sun Devil Stadium is 0 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field? 86) 7
Answer Key Testname: FINREV 1) center (, -); r = 3 10 y -10 10 x ) no solution 3) {- 7 0 } ) {-1, 0, 3} 5) 6) {5} -6 ± 1 3-10 7) x = 3 1, x = - 3 1 8) 8x + h + 5 9) x -7, x -5, x -7 10) x 1 11) 15 1) Odd 13) -x - x + 0 1) 0x + 5x + 10 15) A = x + 108 x 16) (-3, -37); x = -3 17) $6 8
Answer Key Testname: FINREV 18) (a) The x-intercepts are, 3, and. The y-intercept is 88. (b) The graph crosses the x-axis at and, and touches it at 3. (c) The graph resembles f(x) = x6 for large values of x. (d) The graph has 3 turning points. (e) (.57, -0.05), (3.77, -0.76) (f) y 1 - x 19) D 0) P(x) = (x + )(x - 3)(x - 5) 1) x =, x = - ) y = 8 5 3) y = x + 10 ) R(x) = -(x + 1)(x + ), x -, 3 (x + )(x - 3) 5) (a) P(0) = 15 insects (b) P(10) 608 insects (c) 1,350 6) (-«, -6) or (-1, 1) -9-8 -7-6 -5 - -3 - -1 0 1 3 5 6 7 8 9 7) (-«, 1] or [3, «) 8) real zeros of P: - 5, -1, ; P(x) = (x + 5)(x + 1)(x - ) 9) { 1, 10, - 10 } 30) - i, -3 + i 31) f(x) = z3 - z - z + 0 3) 1, -1, - i, + i 33) y = 3 x + 8-3) x = 3 or x = -3 35) ln 15 = x 36) - 37) x > 9 38) {15} 9
Answer Key Testname: FINREV 39) 10.1 hrs 0) {-1} 1) x = 6 ) f(t) = 00,000e0.38t 3) 15,856 ) 75 minutes 5) 18.9 years 6) - 7) 0 8) 3 9) 33 7 + 50) - 1 51) - 1 5) -3 7 7 53) 1 5) 33 55) - 3 56) p 5 57) y = sin 3 px 58) p, 3p 59) - 3p, - p, p, 3p 60) p units to the right 61) 6 y - 3 - x - -6 6) y = sin (x - 3) 10
Answer Key Testname: FINREV 63) - p 6 6) 3p 65) 5p 6 66) 3 5 67) p 3 68) 69) 3 70) - 3 71) 3 5 7) sin a 73) 5 7) ( 3-1) - 1 5 75) + 3 76) p 8, 7p 8, 9p 8, 15p 8 77) x = p, 3p, 5p, 7p 78) q = 1.91,.37 79) x = p 8, 7p 8, 9p 8, 15p 8 80) g = 111e0', b = 9.5, c = 1.66 81) a = 19.7e, g = 18.93e, c = 1.6; a' = 160.53e, g' = 7.87e, c' = 3.87 8) c = 16., a = 0.e, b = 3.5e 83) 196.8 m, 171.7 m 8) 6,73 feet 85) 19.9e; 51.3 miles 86) 36.3 feet 11