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All Textbook Solutions for Calculus Volume 1

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 1. x y x y -3 9 1 1 2 4 2 4 -1 1 3 9 0 0 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 2. x y x y -3 -2 1 1 -2 -8 2 8 -1 -1 3 -2 0 0 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 3. x y x y 1 -3 1 1 2 -2 2 2 3 -1 3 3 0 0 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 4. x y x y 1 1 5 1 2 1 6 1 3 1 7 1 4 1 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 5. x y x y 3 3 15 1 5 2 21 2 8 1 33 3 10 0 For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function. 6. x y x y -7 11 1 -2 -2 5 3 4 -2 1 6 11 0 -1 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) 7. f(x)=5x2 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) f(x)=4x23x+1 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) 9. f(x)=2x For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) 10. f(x)=|x7|+8 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) 11. f(x)=6x+5 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a) f.f(a+h) 12. f(x)=x23x+7 For the following exercises, find the values for each function, if they exist, then simplify. a. f(0) b. f(1) c. f(3) d. f(x) e. f(a)f.f(a+h) 13. f(x)=9 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 14. f(x)=xx216 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 15. g(x)=8x1 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 16. h(x)=3x2+4 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 17. f(x)=1+x+2 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 18. f(x)=1x9 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 19. g(x)=3x4 For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 20. f(x)=4|x+5|For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 21. g(x)=7x5 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 22. f(x)=x2+1 x y x y -3 10 1 2 -2 5 2 5 -1 2 3 10 0 1 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 23. f(x)=3x6 x y x y -3 -15 1 -3 -2 -12 2 0 -1 -9 3 3 0 -6 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 24. f(x)=12x+1 x y x y -3 12 1 32 -2 0 2 2 -1 12 3 52 0 1 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 25. f(x)=2|x| x y x y -3 6 1 2 -2 4 2 4 -1 2 3 6 0 0 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 26. f(x)=x2 x y x y -3 -9 1 -1 -2 -4 2 -4 -1 -1 3 -9 0 0 For the following exercises, set up a table to sketch the graph of each function using the following values: x = -3, -2, -1, 0, 1, 2, 3. 27. f(x)=x3 x y x y -3 -27 1 1 -2 -8 2 8 -1 -1 3 27 0 0 For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph: Domain and range x-intercept, if any (estimate where necessary) y -Intercept, if any (estimate where necessary) The intervals for which the function is increasing The intervals for which the function is decreasing The intervals for which the function is constant Symmetry about any axis and/or the origin Whether the function is even, odd, or neither For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 36. f(x)=3x+4,g(x)=x2 For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 37. f(x)=x8,g(x)=5x2 For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 38. f(x)=3x2+4x+1,g(x)=x=1 For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 39. f(x)=9x2,g(x)=x22x3 For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 40. f(x)=x,g(x)=x2 For the following exercises, for each pair of functions, find a.f+gb.fgc.f.gd.f/g. Determine the domain of each of these new functions. 41. f(x)=6+1x,g(x)=1x For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 42. f(x)=3x,g(x)=x+5 For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 43. f(x)=x+4,g(x)=4x1 For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 44. f(x)=2x+4,g(x)=x22 For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 45. f(x)=x2+7,g(x)=x23 For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 46. f(x)=x,g(x)=x+9 For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 47. f(x)=32x+1,g(x)=2x For the following exercises, for each pair of functions, find a.(fg)(x) and b.(gf)(x) Simplify the results. Find the domain of each of the results. 48. f(x)=|x+1|,g(x)=x2+x4 The table below lists the NBA championship winners for the years 2001 to 2012. Year Winner 2001 LA Lakers 2002 LA Lakers 2003 San Antonio Spurs 2004 Detroit Pistons 2005 San Antonio Spurs 2006 Miami Heat 2007 San Antonio Spurs 2008 Boston Celtics 2009 LA Lakers 2010 LA Lakers 2011 Dallas Mavericks 2012 Miami Heat Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.The area A of a square depends on the length of the side s. Write a function A(s) for the area of a square. Find and interpret A(6.5). Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.[T] The volume of a cube depends on the length of the sides s. Write a function V(s) for the area of a square. Find and interpret V(11.8).[T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost C to rent a car is a function of the hours t the car is rented plus the flat fee. Write the formula for the function that models this situation. Find the total cost to rent a car for 2 days and 7 hours. Determine how long the car was rented if the bill is $432.73.[T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank. Write a formula that models this situation, Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas. Determine the domain and range of the function. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.[T] The volume V of a sphere depends on the length of its radius as V=(4/3)r3. Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius 6.371106m .[T] A certain bacterium grows in culture in a circular region. Tire radius of the circle, measured in centimeters, is given by r(t)=6[5/t2+1] , where f is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture. Express the area of the bacteria as a function of time. Find the exact and approximate area of the bacterial culture in 3 hours. Express the circumference of the bacteria as a function of time. Find the exact and approximate circumference of the bacteria in 3 hours.[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function E(x) = 0.79x, where x is the number of U.S. dollars and E(x) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is D(x) = 1.245x, where x is the number of Euros and D(x) is the equivalent number of U.S. dollars. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process? Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.[T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell. Write a function y = S(x) that models a worker’s monthly salary based on the number of skateboards x he or she sells. Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line y = 1400.)Use a graphing calculator to graph the half-circle y=25( x4)2. Then, use the INTERCEPT feature to find value of both the x- and y-intercepts.For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 59. (2,4)and(1,1)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 60. (1,4)and(3,1)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 61. (3,5)and(1,2)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 62. (6,4)and(4,3)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 63. (2,3)and(5,7)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 64. (1,9)and(8,5)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 65. (2,4)and(1,4)For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. 66. (1,4)and(1,0)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 67. Slope = -6, passes through (1, 3)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 68. Slope = 3, passes through (-3. 2)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 69. Slope = 13 , passes through (0, 4)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 70. Slope = 25 , x -intercept = 8 For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 71. Passing through (2,1)and(2,1)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 72. Passing through (3,7)and(1,2)For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 73. x -intercept = 5 and y -intercept = - 3 For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. 74. x-intercept = -6 and y -intercept = 9 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 75. y=2x3 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 76. y=17x+1 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 77. f(x)=6x For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 78. f(x)=5x+4 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 79. 4y+24=0 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 80. 8x4=0 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 81. 2x+3y=6 For the following exercises, for each linear equation, a. give the slope m and y -intercept b, if any, and b. graph the line. 82. 6x5y+15=0 For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 83. f(x)=2x23x5 For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 84. f(x)=3x2+6x For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 85. f(x)=12x21 For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 86. f(x)=x3+3x2x3 For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. 87. f(x)=3xx3 For the following exercises, use the graph of f(x)=x2 to graph each transformed function g. 88. g(x)=x21 For the following exercises, use the graph of f(x)=x2 to graph each transformed function g. 89. g(x)=(x+3)2+1 For the following exercises, use the graph of f(x)=x2 to graph each transformed function g. 90. g(x)=x+2 For the following exercises, use the graph of f(x)=x2 to graph each transformed function g. 91. g(x)=x1 For the following exercises, use the graph of y=f(x) to graph each transformed function g. 92. g(x)=f(x)+1 For the following exercises, use the graph of y=f(x) to graph each transformed function g. 93. g(x)=f(x1)+2 For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. 94. f(x)={4x+3,x0;f(3);f(0);f(2)x+1,x0 For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. 95. f(x)={x23,x0;f(4);f(0);f(2)4x3,x0 For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. 96. h(x)={x+1,x5;h(0);h();h(5)4,x5 For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. 97. g(x)={3x2,x2;g(0);g(4);g(2)4,x=2 For the following exercises, determine whether the statement is true or false. Explain why. 98. f(x)=(4x+1)/(7x2) is a transcendental function.For the following exercises, determine whether the statement is true or false. Explain why. 99. g(x)=x3 is an odd root function For the following exercises, determine whether the statement is true or false. Explain why. 100. A logarithmic function is an algebraic function.For the following exercises, determine whether the statement is true or false. Explain why. 101. A function of the form f(x)=xb, where b is a real valued constant, is an exponential function.For the following exercises, determine whether the statement is true or false. Explain why. 102. The domain of an even root function is all real numbers.[T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300. Find a function y = V(t) that determines the value V of the equipment at the end of t years. Find and interpret the meaning of the x- and y-intercepts for this situation. What is the value of the equipment at the end of 5 years? When will the value of the equipment be $3000?[T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t = 0), total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t. Interpret the slope of the graph of S. Use part a. to predict the year when online shopping during Christmas will reach $60 billion.[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost C as a function of number of cupcakes made. Find a linear function that relates cost C to x, the number of cupcakes made. Find the cost to bake 160 cupcakes. If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? {Hint-. Use the INTERSECTION function on a calculator to find this number.)[T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years. Find a linear function that models the price P of the house versus the number of years t since the original purchase. Interpret the slope of the graph of P. Find the price of the house 15 years from when it was originally purchased.[T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year. Find a linear function that models the value V of the car after t years. Find and interpret V(4).[T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.[T] The total cost C (in thousands of dollars) to produce a certain item is modeled by the function C(x)=10.50x+28,500 , where x is the number of items produced. Determine the cost to produce 175 items.[T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four- page assignment and about 1.5 hours to type a nine-page assignment. Find the linear function y=N(t) that models this situation, where N is the number of pages typed and t is the time in minutes. Use part a. to determine how many pages can be typed in 2 hours. Use part a. to determine how long it takes to type a 20-page assignment.[T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function P(t)=1.8576t+68.052 , where f is time in years and t = 0 corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.[T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll. Find the linear function y=N(x), where N is the number of students that actually enroll and x is the number of all students offered admission to the class of 2019. If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 113. 240 For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 114. 15 For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 115. 60 For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 116. 225 For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of . 117. 330 For the following exercises, convert each angle in radians to degrees. 118. 2rad For the following exercises, convert each angle in radians to degrees. 119. 76rad For the following exercises, convert each angle in radians to degrees. 120. 112rad For the following exercises, convert each angle in radians to degrees. 121. 3rrad For the following exercises, convert each angle in radians to degrees. 122. 512rad Evaluate the following functional values. 123. cos(43)Evaluate the following functional values. 124. tan(194)Evaluate the following functional values. 125. sin(34)Evaluate the following functional values. 126. sec(6)Evaluate the following functional values. 127. sin(12)Evaluate the following functional values. 128. cos(512)For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 129. a = 4, c = 7 For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle, b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 130. a = 21, c = 29 For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 131. a = 85.3, b = 125.5 For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 132. b = 40, c = 41 For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 133. a = 84, b = 13 For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 134. b = 28, c = 35 For the following exercises, P is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point P. Rationalize denominators. 135. P(725,y),y0 For the following exercises, P is a point on the unit circle, a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point P. Rationalize denominators. 136. P(1517,y),y0 For the following exercises, P is a point on the unit circle, a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point P. Rationalize denominators. 137. P(x,73),x0 For the following exercises, P is a point on the unit circle, a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point P. Rationalize denominators. 138. P(x, 154),x0 For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 139. tan2x+sinxcscx For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 140. secxsinxcotx For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 141. tan2xsec2x For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 142. secxcosx For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 143. (1+tan)22tan For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 144. sinx(cscxsinx)For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 145. costsint+sint1+cost For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. 146. 1+tan21+cot2 For the following exercises, verify that each equation is an identity. 147. tancotcsc=sin For the following exercises, verify that each equation is an identity. 148. sec2tan=seccsc For the following exercises, verify that each equation is an identity. 149. sintcsct+costsect=1 For the following exercises, verify that each equation is an identity. 150. sinxcosx+1+cosx1sinx=0 For the following exercises, verify that each equation is an identity. 151. cot+tan=seccsc For the following exercises, verify that each equation is an identity. 152. sin2+tan2+cos2=sec2 For the following exercises, verify that each equation is an identity. 153. 11sin+11+sin=2sec2 For the following exercises, verify that each equation is an identity. 154. tancotsincos=sec2csc2 For the following exercises, solve the trigonometric equations on the interval 02 . 155. 2sin1=0 For the following exercises, solve the trigonometric equations on the interval 02 . 156. 1+cos=12 For the following exercises, solve the trigonometric equations on the interval 02 . 157. 2tan2=2 For the following exercises, solve the trigonometric equations on the interval 02 . 158. 4sin22=0 For the following exercises, solve the trigonometric equations on the interval 02 . 159. 3cot+1=0 For the following exercises, solve the trigonometric equations on the interval 02 . 160. 3sec23=0 For the following exercises, solve the trigonometric equations on the interval 02 . 161. 2cossin=sin For the following exercises, solve the trigonometric equations on the interval 02 . 162. csc2+2csc+1=0 For the following exercises, each graph is of the form y=AsinBxory=AcosBx , where B >0. Write the equation of the graph. 163.For the following exercises, each graph is of the form y=AsinBxory=AcosBx , where B >0. Write the equation of the graph. 164.For the following exercises, each graph is of the form y=AsinBxory=AcosBx , where B >0. Write the equation of the graph. 165.For the following exercises, each graph is of the form y=AsinBxory=AcosBx , where B >0. Write the equation of the graph. 166.For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 167. y=sin(x4)For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 168. y=3cos(2x+3)For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 169. y=12sin(14x)For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 170. y=2cos(x3)For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 171. y=3sin(x+2)For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function, 172. y=4cos(2x2)[T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of 120°, how many inches does it move? Approximate to the nearest whole inch.[T] Find the length of the arc intercepted by central angle 3 in a circle of radius r. Round to the nearest hundredth. a.r=12.8cm,=56radb.r=4.378cm,=76radc.r=0.964cm,=50d.r=8.55cm,=325 [T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, , and is given by =/t. where is in radians and t is time. Find the angular speed for die given data. Round to the nearest thousandth. a. =74rad,t=10sec b. =35rad,t=8sec c. =29rad,t=1min d. =23.76rad,t=14min [T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular. Find the radius of the circular land area. If the land area is to form a 45° sector of a circle instead of a whole circle, find the length of the curved side.[T] The area of an isosceles triangle with equal sides of length x is 12x2sin , where is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle =5/12rad .[T] A panicle travels in a circular path at a constant angular speed . The angular speed is modeled by the function =9|cos(t/12)| . Determine the angular speed at t = 9 sec.[T] An alternating current for outlets in a home has voltage given by the function V(t)=150cos368t , where V is the voltage in volts at time t in seconds. Find the period of the function and interpret its meaning. Determine the number of periods that occur when 1 sec has passed.[T] The number of hours of daylight in a northeast city is modeled by the function N(t)=12+3sin[2365(t79)] , where t is the number of days after January 1. Find the amplitude and period. Determine the number of hours of daylight on the longest day of the year. Determine the number of hours of daylight on the shortest day of the year. Determine the number of hours of daylight 90 days after January 1. Sketch the graph of the function for one period starting on January 1.[T] Suppose that T=50+10sin[12(t8)] is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week. Determine the amplitude and period. Find the temperature 7 hours after midnight. At what time does T = 60°? Sketch the graph of T over 0t24 .[T] The function H(t)=8sin(6t) models the height H (in feet) of the tide f hours after midnight. Assume that t = 0 is midnight. Find the amplitude and period. Graph the function over one period. What is the height of the tide at 4:30 a.m.?Consider the graph in Figure 1.42 of the function y=sinx+cosx . Describe its overall shape. Is it periodic? How do you know? Figure 1.42 The graph of y=sinx+cosx . Using a graphing calculator or other graphing device, estimate the x - and y -values of the maximum point for the graph (the first such point where x >0). It may be helpful to express the x -value as a multiple of .Now consider other graphs of the form y=Asinx+Bcosx for various values of A and B. Sketch the graph when A = 2 and B = 1, and find the x - and y-values for the maximum point. (Remember to express the x-value as a multiple of , if possible.) Has it moved?Repeat for A = 1, B = 2. Is there any relationship to what you found in part (2)?Complete the following table, adding a few choices of your own for A and B:5. Try to figure out the formula for the y-values.6. The formula for the x-values is a little harder. The most helpful points from the table are (1,1),(1,3),(3,1) . (Hint: Consider inverse trigonometric functions.)7. If you found formulas for parts (5) and (6), show that they work together. That is, substitute the x -value formula you found into y=Asinx+Bcosx and simplify it to arrive at the y -value formula you found.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 189. f(x)=x24,x0 For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 190. f(x)=x43 For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 191. f(x)=x3+1 For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 192. f(x)=(x1)2,x1 For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 193. f(x)=x1 For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. 194. f(x)=1x+2 For the following exercises, use the graph of f sketch the graph of its inverse function. 195.For the following exercises, use the graph of f sketch the graph of its inverse function. 196.For the following exercises, use the graph of f sketch the graph of its inverse function. 197.For the following exercises, use the graph of f to sketch the graph of its inverse function. 198.For the following exercises, use composition to determine which pairs of functions are inverses. 199. f(x)=8x,g(x)=x8 For the following exercises, use composition to determine which pairs of functions are inverses. 200. f(x)=8x+3,g(x)=x38 For the following exercises, use composition to determine which pairs of functions are inverses. 201. f(x)=5x7,g(x)=x+57 For the following exercises, use composition to determine which pairs of functions are inverses. 202. f(x)=23x+2,g(x)=32x+3 For the following exercises, use composition to determine which pairs of functions are inverses. 203. f(x)=1x1,x1,g(x)=1x+1,x0 For the following exercises, use composition to determine which pairs of functions are inverses. 204. f(x)=x3+1,g(x)=(x1)1/3 For the following exercises, use composition to determine which pairs of functions are inverses. 205. f(x)=x2+2x+1,x1,g(x)=1+x,x0 For the following exercises, use composition to determine which pairs of functions are inverses. 206. f(x)=4x2,0x2,g(x)=4x2,0x2 For the following exercises, evaluate the functions. Give the exact value. 207. tan1(33)For the following exercises, evaluate the functions. Give the exact value. 208. cos1(22)For the following exercises, evaluate the functions. Give the exact value. 209. cot1(1)For the following exercises, evaluate the functions. Give the exact value. 210. sin1(1)For the following exercises, evaluate the functions. Give the exact value. 211. cos1(32)For the following exercises, evaluate the functions. Give the exact value. 212. cos(tan1(3))For the following exercises, evaluate the functions. Give the exact value. 213. sin(cos1( 2 2))For the following exercises, evaluate the functions. Give the exact value. 214. sin1(sin(3))For the following exercises, evaluate the functions. Give the exact value. 215. tan1(tan(6))The function C=T(F)=(5/9)(F32) converts degrees Fahrenheit to degrees Celsius. Find the inverse function F=T1(C) What is the inverse function used for?[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function V=f(x)=500(0.04x2) for 0x0.2 . Find x=f1(V) Interpret what the inverse function is used for. Find the distance from the center of an artery with a velocity of 15 cm. sec, 10 cm/sec, and 5 cm/sec.A function that converts dress sizes in the United States to those in Europe is given by D(x)=2x+24 . Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States. Find the function that converts European dress sizes to U.S. dress sizes. Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.[T] The cost to remove a toxin from a lake is modeled by the function C(p)=75p/(85p) , where C is the cost (in thousands of dollars) and p is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake. Find the inverse function, c. Use pan b. to determine how much of the toxin is removed for $50,000.[T] A race car is accelerating at a velocity given by v(t)=254t+54 , where v is the velocity (in feet per second) at time t. Find the velocity of the car at 10 sec. Find the inverse function. Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.[T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by =2sin1(1M) . Find the Mach angle (to the nearest degree) for the following numbers. a.=1.4b.=2.8c.=4.3 [T] Using =2sin1(1M) , find the Mach number M for the following angles. a.=6b.=27c.=38 [T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function T(x)=5+18sin[6(x4.6)] , where x is time in months and x = 1.00 corresponds to January 1. Determine the month and day when the temperature is 21 °C.[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function D(x)=5sin(6t76)+8 , where t is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.[T] All object moving in simple harmonic motion is modeled by the function s(t)=6cos(t2) , where s is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4.5 in.A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle can be modeled by the function =tan15.5xtan12.5x, where x is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.[T] Use a calculator to evaluate tan1(tan(2.1))andcos-1(cos(2.1)) . Explain the results of each.[T] Use a calculator to evaluate sin(sin1(2))andtan(tan1(2)) . Explain the results of each.For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. 229. f(x)=5xa.x=3b.x=12c.x=2 For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. 230. f(x)=(0.3)xa.x=1b.x=4c.x=1.5 For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. 231. f(x)=10xa.x=2b.x=4c.x=53 For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal. 232. f(x)=exa.x=2b.x=3.2c.x=For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 233.For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 234.For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 235.For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 236.For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 237.For the following exercises, match the exponential equation to the correct graph. y=4x y=3x1 y=2x+1 y=(12)x+2 y=3x y=15x 238.For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 239. f(x)=ex+2 For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 240. f(x)=2x For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 241. f(x)=3x+1 For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 242. f(x)=4x1 For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 243. f(x)=12x For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 244. f(x)=5x+1+2 For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote. 245. f(x)=ex1 For the following exercises, write the equation in equivalent exponential form. 246. log381=4 For the following exercises, write the equation in equivalent exponential form. 247. log82=13 For the following exercises, write the equation in equivalent exponential form. 248. log51=0 For the following exercises, write the equation in equivalent exponential form. 249. log525=2 For the following exercises, write the equation in equivalent exponential form. 250. log0.1=1 For the following exercises, write the equation in equivalent exponential form. 251. In(1e3)=3 For the following exercises, write the equation in equivalent exponential form. 252. log93=0.5 For the following exercises, write the equation in equivalent exponential form. 253. In1=0 For the following exercises, write the equation in equivalent logarithmic form. 254. 23=8 For the following exercises, write the equation in equivalent logarithmic form. 255. 42=116 For the following exercises, write the equation in equivalent logarithmic form. 256. 102=100 For the following exercises, write the equation in equivalent logarithmic form. 257. 90=1 For the following exercises, write the equation in equivalent logarithmic form. 258. (13)3=127 For the following exercises, write the equation in equivalent logarithmic form. 259. 643=4 For the following exercises, write the equation in equivalent logarithmic form. 260. ex=y For the following exercises, write the equation in equivalent logarithmic form. 261. 9y=150 For the following exercises, write the equation in equivalent logarithmic form. 262. b3=45 For the following exercises, write the equation in equivalent logarithmic form. 263. 43/2=0.125 For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 264. f(x)=3+Inx For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 265. f(x)=In(x1)For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 266. f(x)=In(x)For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 267. f(x)=1Inx For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 268. f(x)=logx1 For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. 269. f(x)=In(x+1)For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 270. logx4y For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 271. log39a3b For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 272. Inab3 For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 273. log5125xy3 For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 274. log4xy364 For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and or product of logarithms. 275. In(6 e 3 )For the following exercises, solve the exponential equation exactly. 276. 5x=125 For the following exercises, solve the exponential equation exactly. 277. e3x15=0 For the following exercises, solve the exponential equation exactly. 278. 8x=4 For the following exercises, solve the exponential equation exactly. 279. 4x+132=0 For the following exercises, solve the exponential equation exactly. 280. 3x/14=110 For the following exercises, solve the exponential equation exactly. 281. 10x=7.21 For the following exercises, solve the exponential equation exactly. 282. 423x20=0 For the following exercises, solve the exponential equation exactly. 283. 73x2=11 For the following exercises, solve the logarithmic equation exactly, if possible. 284. log3x=0 For the following exercises, solve the logarithmic equation exactly, if possible. 285. log5x=2 For the following exercises, solve the logarithmic equation exactly, if possible. 286. log4(x+5)=0 For the following exercises, solve the logarithmic equation exactly, if possible. 287. log(2x7)=0 For the following exercises, solve the logarithmic equation exactly, if possible. 288. Inx+3=2 For the following exercises, solve the logarithmic equation exactly, if possible. 289. log6(x+9)+log6x=2 For the following exercises, solve the logarithmic equation exactly, if possible. 290. log4(x+2)log4(x1)=0 For the following exercises, solve the logarithmic equation exactly, if possible. 291. Inx+In(x2)=In4 For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 292. log547 For the following exercises, rise the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. 293. log782

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