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All Textbook Solutions for Calculus Volume 1

For the following functions y=f(x) , find f(a) using Equation 3.3. 21. f(x)=5x+4,a=1 For the following functions y=f(x) , find f(a) using Equation 3.3. 22. f(x)=7x+1,a=3 For the following functions y=f(x) , find f(a) using Equation 3.3. 23. f(x)=x2+9x,a=2 For the following functions y=f(x) , find f(a) using Equation 3.3. 24. f(x)=3x2x+2,a=1 For the following functions y=f(x) , find f(a) using Equation 3.3. 25. f(x)=x,a=4 For the following functions y=f(x) , find f(a) using Equation 3.3. 26. f(x)=x2,a=6 For the following functions y=f(x) , find f(a) using Equation 3.3. 27. f(x)=1x,a=2 For the following functions y=f(x) , find f(a) using Equation 3.3. 28. f(x)=1x3,a=1 For the following functions y=f(x) , find f(a) using Equation 3.3. 29. f(x)=1x3,a=1 For the following functions y=f(x) , find f(a) using Equation 3.3. 30. f(x)=1x,a=4 For the following exercises, given the function y=f(x), find the slope of the secant line PQ for each point Q(x,f(x)) with value given in the table. Use the answers from a. to estimate the value of the slope of the tangent line at P. Use the answer from b. to find the equation of the tangent line to f at point P, 31. [T] f(x)=x2+3x+4, P(1, 8) (Round to 6 decimal places.) X Slope mPQ X Slope mPQ 1.1 (i) 0.9 (vh) 1.01 (ii) 0.99 (viii) 1.001 (hi) 0.999 (ix) 1.0001 (iv) 0.9999 (x) 1.00001 (v) 0.99999 (xi) 1.000001 (vi) 0.999999 (xii)For the following exercises, given the function y=f(x), find the slope of the secant line PQ for each point Q(x,f(x))with x value given in the table. Use the answers from a. to estimate the value of the slope of the tangent line at P. Use the answer from b. to find the equation of the tangent line to f at point P. 32. [T] f(x)=x+1x21,P(0,1) x Slope mPQ x Slope mPQ 0.1 (i) -0.1 (vii) 0.01 (ii) -0.01 (viii) 0.001 (iii) -0.001 (ix) 0.0001 (iv) -0.0001 (X) 0.00001 (v) -0.00001 (xi) 0.000001 (Vi) -0.000001 (xii)For the following exercises, given the function y=f(x) , find the slope of the secant line PQ for each point Q(x,f(x))with x value given in the table. Use the answers from a. to estimate the value of the slope of the tangent line at P. Use the answer from b. to find the equation of the tangent line to f at point P. 33. [T] f(x)=10e0.5x,P(0,10) (Round to 4 decimal places.) x Slope mPQ -0.1 (i) -0.01 (ii) -0.001 (iii) -0.0001 (iv) -0.00001 (v) -0.000001 (vi)For the following exercises, given the function y=f(x) , find the slope of the secant line PQ for each point Q(x,f(x))with x value given in the table. Use the answers from a. to estimate the value of the slope of the tangent line at P. Use the answer from b. to find the equation of the tangent line to f at point P. 34. [T] f(x)=tan(x),P(,0) (Round to 4 decimal places.) x Slope mPQ 3.1 (i) 3.14 (ii) 3.141 (iii) 3.1415 (iv) 3.14159 (v) 3.141592 (vi)[T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 35. s(t)=13t+5 [T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 36. s(t)=t22t [T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 37. s(t)=2t3+3 [T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 38. s(t)=16t24t [T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 39. Use the following graph to evaluate a. f’(1) and b. f’(6).[T] For the following position functions y=s(t), an object is moving along a straight line, where t is in seconds and s is in meters. Find the simplified expression for the average velocity from t=2 to t=2+h the average velocity between t=2 and t=2+h, where (i) h = 0.1, (ii) h = 0.01, (iii)h = 0.001, and (iv) h = 0.0001; and use the answer from a. to estimate the instantaneous velocity at t=2 second. 40. Use the following graph to evaluate a. f’(3) and b. f’(1.5).For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x=a for each of the given functions. 41. f(x)=x1/3,x=0 For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x=a for each of the given functions. 42. f(x)=x2/3,x=0 For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x=a for each of the given functions. 43. f(x)={1,x1x,x1,x=1 For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x=a for each of the given functions. 44. f(x)=|x|x,x=0 [T] The position in feet of a lace car along a straight track after t seconds is modeled by the function s(t)=8t2116t3 . a.Find the average velocity of the vehicle over the following time intervals to four decimal places: [4,4.1] [4,4.01] [4,4.001] [4,4.0001] b.Use a. to draw a conclusion about the instantaneous velocity of the vehicle at t = 4 seconds.[T] The distance in feet that a ball rolls down an incline is modeled by the function s(t)=14t2 , where t is seconds after the ball begins rolling. a.Find the average velocity of the ball over the following time intervals: [5,5.1] [5, 5.01] [5, 5.001] [5, 5.0001] b.Use the answers from a. to draw a conclusion about the instantaneous velocity of die ball at t = 5 seconds.Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by s=f(t) and s=g(t) , where s is measured in feet and t is measured in seconds. Which vehicle has traveled farther at t = 2 seconds? What is the approximate velocity of each vehicle at t = 3 seconds? Which vehicle is traveling faster at t = 4 seconds? What is true about the positions of the vehicles at t = 4 seconds?[T] The total cost C(x), in hundreds of dollars, to produce x jars of mayonnaise is given by C(x)=0.000003x3+4x+300 . a. Calculate the average cost per jar over the following intervals: i. [100, 100.1] ii. [100, 100.01] iii. [100, 100.001] iv. [100, 100.0001] b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.[T] For the function f(x)=x32x211x+12 , do the following. Use a graphing calculator to graph f in an appropriate viewing window. Use die ZOOM feature on the calculator to approximate the two values of x=a for which mtan=f(a)=0 .[T] For the function f(x)=x1+x2 , do the following. Use a graphing calculator to graph f in an appropriate viewing window. Use die ZOOM feature on the calculator to approximate the values of x = a for which mtan=f(a)0 .Suppose that N(x) computes the number of gallons of gas used by a vehicle traveling x miles. Suppose the vehicle gets 30 mpg. Find a mathematical expression for N(x). What is N( 100)? Explain the physical meaning. What is N’(100)? Explain the physical meaning.[T] For the function f(x)=x45x2+4 , do the following. Use a graphing calculator to graph f in an appropriate viewing window. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate f’(-2), f'(—0.5), f’(1.7), and f’(2.718).[T] For the function f(x)=x2x2+1 , do the following. Use a graphing calculator to graph f in an appropriate viewing window. Use the nDeriv function on a graphing calculator to find f’(-4), f'(-2), f'(2), and f'(4).For the following exercises, use the definition of a 65. Derivative to find f(x) . 54. f(x)=6 For the following exercises, use the definition of a derivative to find f(x) . 55. f(x)=23x For the following exercises, use the definition of a derivative to find f(x) . 56. f(x)=2x7+1 For the following exercises, use the definition of a derivative to find f(x) . 57. f(x)=4x2 For the following exercises, use the definition of a derivative to find f(x) . 58. f(x)=5xx2 For the following exercises, use the definition of a derivative to find f(x) . 59. f(x)=2x For the following exercises, use the definition of a derivative to find f(x) . 60. f(x)=x6 For the following exercises, use the definition of a derivative to find f(x) 61. f(x)=9x For the following exercises, use the definition of a derivative to find f(x) 62. f(x)=x+1x For the following exercises, use the definition of a derivative to find f(x) 63. f(x)=1x For the following exercises, use the graph of y=f(x) to sketch the graph of its derivative f(x) 64.For the following exercises, use the graph of y=f(x) to sketch the graph of its derivative f(x) 65.For the following exercises, use the graph of y=f(x) to sketch the graph of its derivative f(x) 66.For the following exercises, use the graph of y=f(x) to sketch the graph of its derivative f(x) 67.For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 68. limh0( 1+h)2/31h For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 69. limh0[3( 2+h)2+2]14h For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 70. limh0cos(+h)+1h For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 71. limh0( 2+h)416h For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 72. limh0[2( 3+h)2(3+h)]15h For the following exercises, the given limit represents the derivative of a function y=f(x) at x = a. Find f(x) and a. 73. limh0eh1h For the following functions, sketch the graph and use the definition of a derivative to show that the function is not differentiable at x=1 . 74. f(x)={2x,0x13x1,x1 For the following functions, sketch the graph and use the definition of a derivative to show that the function is not differentiable at x=1 . 75. f(x)={3,x13x,x1 For the following functions, sketch the graph and use the definition of a derivative to show that the function is not differentiable at x=1 . 76. f(x)={x2+2,x1x,x1 For the following functions, sketch the graph and use the definition of a derivative to show that the function is not differentiable at x=1 . 77. f(x)={2x,x12x,x1 For the following graphs, a. determine for which values of x=a the limxaf(x) exists but f is not continuous at x=a , and b. determine for which values of x=a the function is continuous but not differentiable at x=a.For the following graphs, a. determine for which values of x=a the limxaf(x) exists but f is not continuous at x=a , and b. determine for which values of x=a the function is continuous but not differentiable at x=a. 79.Use the graph to evaluate a. f’(0.5), b. f’(0), c. f’(1), d. f’(2), and e. f’(3), if it exists.For the following functions, use f(x)=limh0f(x+h)f(x)h to find f(x) . 81. f(x)=23x For the following functions, use f(x)=limh0f(x+h)f(x)h to find f(x) . 82. f(x)=4x2 For the following functions, use f(x)=limh0f(x+h)f(x)h to find f(x) . 83. f(x)=x+1x For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 84. [T] f(x)=5x For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 85. [T] f(x)=3x2+2x+4 For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 86. [T] f(x)=x+3x For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 87. [T] f(x)=12x For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 88. [T] f(x)=1+x+1x For the following exercises, use a calculator to graph f(x). Determine the function f(x) , then use a calculator to graph f(x) . 89. [T] f(x)=x3+1 For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 90. P(x) denotes the population of a city at time x in years.For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 91. C(x) denotes the total amount of money (in thousands of dollars) spent on concessions by x customers at an amusement park.For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 92. R(x) denotes the total cost (in thousands of dollars) of manufacturing x clock radios.For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 93. g(x) denotes the grade (in percentage points) received on a test, given x hours of studying.For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 94. B(x) denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in x years since 1990.For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to in elude units. a. f(x+h)f(x)h b. f(x)=limh0f(x+h)f(x)h 95. p(x) denotes atmospheric pressure at an altitude of x feet.Sketch the graph of a function y=f(x) with all of the following properties: f(x)0 for 2x1 f(2)=0 f(x)0 for x2 f(2)=2 and f(0)=1 limxf(x)=0 and limxf(x)= f(1)does not exist.Suppose temperature T in degrees Fahrenheit at a height x in feet above the ground is given by y=T(x). Give a physical interpretation, with units, of T'(x). If we know that T(1000)=0.1 , explain the physical meaning.Suppose the total profit of a company is y=P(x) thousand dollars when x units of an item are sold. What does P(b)P(a)ba for 0 < a < b measure and what are the units? What does P'(x) measure, and what are die units? Suppose that P' (30) = 5, what is the approximate change in profit if the number of items sold increases from 30 to 31 ?The graph in the following figure models die number of people N(t) who have come down with the flu t weeks after its initial outbreak in a town with a population of 50.000 citizens. Describe what N'(t) represents and how it behaves as t increases. What does the derivative tell us about how this town is affected by the flu outbreak?For the following exercises, use the following table, which show’s the height h of the Saturn V rocket for the Apollo 11 mission t seconds after launch. Time (seconds) Height (meters) 0 0 1 2 2 4 3 13 4 25 5 32 100. What is the physical meaning of h(t) ? What are the units?[T] Construct a table of values for h' (t) and graph both h(t) and h’(t) on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them.)[T] The best linear fit to the data is given by H(t)=7.229t4.905 , where H is the height of the rocket (in meters) and t is the time elapsed since takeoff. From this equation, determine H' (t), Graph H(t) with the given data and, on a separate coordinate plane, graph H’(t).[T] The best quadratic fit to the data is given by G(t)=1.429t2+0.0857t0.1429 . where G is the height of the rocket (in meters) and t is the time elapsed since takeoff. From this equation, determine G' (t). Graph G(t) with the given data and, on a separate coordinate plane, graph G' (t).[T] The best cubic fir to the data is given by F(t)=0.2037t3+2.956t22.705t+0.4683 , where F is the height of the rocket (in m) and t is the time elapsed since take off. From this equation, determine F’(t). Graph F(t) with the given data and. on a separate coordinate plane, graph F' (t). Does the linear, quadratic, or cubic function fit die data best?Using the best linear, quadratic, and cubic fits to the data, determine what H(t) , G(t) and F(t) are. What are the physical meanings of H(t) , G(t) and F(t) ,and what are their units?Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the (x, y) coordinates of this point near the turn.Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . 2. Find the equation of the tangent line to the curve at this point.Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . 3. To determine whether the spectators are in danger in this scenario, find the x-coordinates of the point where the tangent line crosses the line y = 2.8. Is this point safely to the right of the grandstand? Or are the spectators in danger?Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . 4. What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point (-2.5, 0.625). What is the slope of the tangent line at this point?Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . 5. If a driver loses control as described in part 4, are the spectators safe?Figure 3.21 (a) one section of the racetrack can be modeled by the function f(x)=x3+3x2+x . (b) The front corner of the grandstand is located at (1.9,2.8) . 6. Should you proceed with the current design for the grandstand, or should the grandstands be moved?For the following exercises, find f(x) for each function. 106. f(x)=x7+10 For the following exercises, find f(x) for each function. 107. f(x)=5x3x+1 For the following exercises, find f(x) for each function. 108. f(x)=4x27x For the following exercises, find f(x) for each function. 109. f(x)=8x4+9x21 For the following exercises, find f(x) for each function. 110. f(x)=x4+2x For the following exercises, find f(x) for each function. 111. f(x)=3x(18x4+13x+1)For the following exercises, find f(x) for each function. 112. f(x)=(x+2)(2x23)For the following exercises, find f(x) for each function. 113. f(x)=x2(2x2+5x3)For the following exercises, find f(x) for each function. 114. f(x)=x3+2x243 For the following exercises, find f(x) for each function. 115. f(x)=4x32x+1x2 For the following exercises, find f(x) for each function. 116. f(x)=x2+4x24 For the following exercises, find f(x) for each function. 117. f(x)=x+9x27x+1 For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 118. [T] y=3x2+4x+1at(0,1)For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 119. [T] y=2x+1at(4,5)For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 120. [T] y=2xx1at(1,1)For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 121. [T] y=2x3x2at(1,1)For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 122. h(x)=4f(x)+g(x)7 For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 123. h(x)=x3f(x)For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 124. h(x)=f(x)g(x)2 For the following exercises, find the equation of the tangent line T(x) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 125. h(x)=3f(x)g(x)+2 For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. x 1 2 3 4 f(x) 3 5 -2 0 8(x) 2 3 -4 6 f’(x) -1 7 8 -3 g'(x) 4 1 2 9 \as 126. Find h(1) if h(x)=xf(x)+4g(x)For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. x 1 2 3 4 f(x) 3 5 -2 0 8(x) 2 3 -4 6 f’(x) -1 7 8 -3 g'(x) 4 1 2 9 \ 127. Find h(2) if h(x)=f(x)g(x) .For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. x 1 2 3 4 f(x) 3 5 -2 0 8(x) 2 3 -4 6 f’(x) -1 7 8 -3 g'(x) 4 1 2 9 \ 128. Find h(3) if h(x)=2x+f(x)g(x) .For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. x 1 2 3 4 f(x) 3 5 -2 0 8(x) 2 3 -4 6 f’(x) -1 7 8 -3 g'(x) 4 1 2 9 129. Find h(4) if h(x)=1x+g(x)f(x) .For the following exercises, use the following figure to find the indicated derivatives, if they exist. 130. Let h(x)=f(x)+g(x) . Find a. h(1) , b. h(3) , and c. h(4) .For the following exercises, use the following figure to find the indicated derivatives, if they exist. 131. Let h(x)=f(x)g(x) . Find a. h(1) , b. h(3) , and c. h(4) .For the following exercises, use the following figure to find the indicated derivatives, if they exist. 132. Let h(x)=f(x)g(x) . Find a. h(1) , b. h(3) , and c. h(4) .For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 133. [T] f(x)=2x3+3xx2,a=2 For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 134. [T] f(x)=1xx2,a=1 For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 135. [T] f(x)=x2x12+3x+2,a=0 For the following exercises, a. evaluate f(a), and b. graph the function f(x) and the tangent line at x = a. f(x)=1xx2/3,a=1 For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 137. Find the equation of the tangent line to the graph of f(x)=2x3+4x25x3atx=1 For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 138. Find the equation of the tangent line to the graph of f(x)=x2+4x10atx=8 For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 139. Find the equation of the tangent line to the graph of f(x)=(3xx2)(3xx2) at x = 1.For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 140. Find the equation of the tangent line to the graph of f(x)=x3 such that the tangent line at that point has an x intercept of 6.For the following exercises, evaluate f(a), and graph the function f(x) and the tangent line at x = a. 141. Find the equation of the line passing through the point P(3, 3) and tangent to the graph of f(x)=6x1.Determine all points on the graph of f(x)=x3+x2x1 for which the slope of the tangent line is horizontal -1.Find a quadratic polynomial such that f(1)=5 , f(1)=3 and f(1)=6 .A car driving along a freeway with traffic has traveled s(t)=t36t2+9t meters in t seconds. Determine the time in seconds when the velocity of the car is 0. Determine the acceleration of the car when the velocity is 0.[T] A herring swimming along a straight line has traveled s(t)=t2t2+2 feet in t seconds. Determine the velocity of the herring when it has traveled 3 seconds.The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function P(t)=8t+30.2t2+1 , where t is measured in years. Determine the initial flounder population. Determine P' (10) and briefly interpret the result.[T] The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C(t)=2t2+tt3+50 milligrams per liter of blood. Find the rate of change of C(t). Determine the rate of change for t = 8, 12, 24, and 36. Briefly describe what seems to be occurring as the number of hours increases.A book publisher has a cost function given by C(x)=x3+2x+3x2 , where x is the number of copies of a book in thousands and C is the cost, per book, measured in dollars. Evaluate C' (2) and explain its meaning.149. [T] According to Newton’s law of universal gravitation, the force F between two bodies of constant mass m1 and m2 is given by the formula F=Gm1m2d2 , where G is the gravitational constant and d is the distance between the bodies. Suppose that G, m1 , and m2 are constants. Find the rate of change of force F with respect to distance d. Find the rate of change of force F with gravitational constant G=6.671011Nm2/kg2 on two bodies 10 meters apart, each with a mass of 1000 kilograms.For the following exercises, the given functions represent the position of a particle traveling along a horizontal line. Find the velocity and acceleration functions. Determine the time intervals when the object is slowing down or speeding up. 150. s(t)=2t33t212t+8 For the following exercises, the given functions represent the position of a particle traveling along a horizontal line. Find the velocity and acceleration functions. Determine the time intervals when the object is slowing down or speeding up. 151. s(t)=2t315t2+36t10 For the following exercises, the given functions represent the position of a particle traveling along a horizontal line. Find the velocity and acceleration functions. Determine the time intervals when the object is slowing down or speeding up. 152. s(t)=t1+t2 A rocket is fired vertically upward from the ground. The distance s in feet that the rocket travels from the ground after t seconds is given by s(t)=16t2+560t . Find the velocity of the rocket 3 seconds after being fired. Find the acceleration of the rocket 3 seconds after being fired.A ball is thrown downward with a speed of 8 ft/s from the top of a 64-foot-tall building. After t seconds, its height above the ground is given by s(t)=16t28t+64 . Determine how long it takes for the ball to hit the ground. Determine the velocity of the ball when it hits the ground.The position function s(t)=t23t4 represents the position of the back of a car backing out of a driveway and then driving in a straight line, where s is in feet and t is in seconds. In this case, s(t)=0 represents the time at which the back of the car is at the garage door, so s(0)=4 is the starting position of the car, 4 feet inside the garage. Determine the velocity of the car when s(t)=0 . Determine the velocity of the car when s(t)=14 The position of a hummingbird flying along a straight line in t seconds is given by s(t)=3t37t meters. Determine the velocity of the bird at t=1 sec. Determine the acceleration of the bird at t=1 sec. Determine the acceleration of the bird when the velocity equals 0.A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after t seconds is given by s(t)=16t2+100t+85 . Find the velocity of the potato after 0.5 s and 5.75 s. Find the speed of the potato at 0.5 s and 5.75 s. Determine when the potato reaches its maximum height. Find the acceleration of the potato at 0.5 s and 1.5 s. Determine how long the potato is in the air. Determine the velocity of the potato upon hitting the ground.The position function s(t)=t38t gives the position in miles of a freight train where east is the positive direction and t is measured in hours. the direction the train is traveling when s(t)=0 Determine the direction the train is traveling when a(t)=0 Determine the time intervals when the train is slowing down or speeding up.The following graph shows die position y=s(t) of an object moving along a straight line. a. Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero. Sketch the graph of the velocity function. Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero. Determine the time intervals when the object is speeding up or slowing down.The cost function, in dollars, of a company that manufactures food processors is given by C(x)=200+7x+x27 , where x is the number of food processors manufactured. Find the marginal cost function. Find the marginal cost of manufacturing 12 food processors. Find the actual cost of manufacturing the thirteenth food processor.The price p (in dollars) and the demand x for a certain digital clock radio is given by the price—demand function p=100.001x . Find the revenue function R(x) . Find the marginal revenue function. Find the marginal revenue at x = 2000 and 5000.[T] A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by P(x)=30x0.3x2250 , where x is the number of skateboards sold. Find the exact profit from the sale of the thirtieth skateboard. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.[T] In general, the profit function is the difference between the revenue and cost functions: P(x)=R(x)C(x) . Suppose the price-demand and cost functions for the production of cordless drills is given respectively by P=1430.03x and C(x)=75,000+65x , where x is the number of cordless drills that are sold at a price of p dollar’s per drill and C(x) is the cost of producing x cordless drills. Find the marginal cost function. Find the revenue and marginal revenue functions. Find R(1000) and R(4000) . Interpret the results. Find the profit and marginal profit functions. Find P(1000) and P(4000) . Interpret the results.A small town in Ohio commissioned an actuarial film to conduct a study that modeled the rate of change of the town's population. The study found that the town’s population (measured in thousands of people) can be modeled by the function P(t)=13t3+64t+3000 , where t is measured in years. Find the rate of change function P' (t) of the population function. Find P '(1), P'(2), P’(3) and P' (4). Interpret what the results mean for the town. Find P"(l), P” (2), P"(3), and P”(4). Interpret what the results mean for the town’s population.[T] A culture of bacteria grows in number according to the function N(t)=3000(1+4tt2+100) where t is measured in hours. Find the rate of change of the number of bacteria. Find N’(0), N’ (10), N’ (20), and N’ (30). Interpret the results in (b). Find N”(0), N"(10), N”(20), and N”(30). Interpret what the answers imply about the bacteria population growth.The centripetal force of an object of mass m is given by F(r)=mv2r where v is the speed of rotation and r is the distance from the center of rotation. Find the rate of change of centripetal force with respect to the distance from the center of rotation. Find the rate of change of centripetal force of an object with mass 1000 kilograms, velocity of 13.89 m/s, and a distance from the center of rotation of 200 meters.The following questions concern the population (in millions) of London by decade in the 19th century, which is listed in the following table. Years since 1800 Population (millions) 1 0.8795 11 1.040 21 1.264 31 1.516 41 1.661 51 2.000 61 2.634 71 3.272 81 3.911 91 4.422 Table 3.5 Population of London Source: http://en.wikipedia.org/wiki/ Demographics_of_London. 167. [T] Using a calculator or a computer program, find the best-fit linear function to measure the population. Find the derivative of the equation in a. and explain its physical meaning. Find the second derivative of the equation and explain its physical meaning.The following questions concern the population (in millions) of London by decade in the 19th century, which is listed in the following table. Years since 1800 Population (millions) 1 0.8795 11 1.040 21 1.264 31 1.516 41 1.661 51 2.000 61 2.634 71 3.272 81 3.911 91 4.422 Table 3.5 Population of London Source: http://en.wikipedia.org/wiki/ Demographics_of_London. 168. [T] Using a calculator or a computer program, find the best-fit quadratic curve through the data. Find the derivative of the equation and explain its physical meaning. Find the second derivative of the equation and explain its physical meaning.For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table. Time after dropping (s) Position (m) 0 0 1 -1 2 -2 3 -5 4 -7 5 -14 169. [T] Using a calculator or computer program, find the best-fit quadratic curve to the data. Find the derivative of the position function and explain its physical meaning. Find the second derivative of the position function and explain its physical meaning.For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table. Time after dropping (s) Position (m) 0 0 1 -1 2 -2 3 -5 4 -7 5 -14 170. [T] Using a calculator or computer program, find the best-fit cubic curve to the data. Find the derivative of the position function and explain its physical meaning. Find the second derivative of the position function and explain its physical meaning. Using the result from c. explain why a cubic function is not a good choice for this problem.For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table. Time after dropping (s) Position (m) 0 0 1 -1 2 -2 3 -5 4 -7 5 -14 The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption. 171. [T] The Holling type I equation is described by f(x)=ax, where x is the amount of prey available and a >0 is the rate at which the predator meets the prey for consumption. Graph the Holling type I equation, given a = 0.5. Determine the first derivative of the Holling type I equation and explain physically what the derivative implies. Determine the second derivative of the Holling type I equation and explain physically what the derivative implies. Using the interpretations from b. and c. explain why the Holling type I equation may not be realistic.For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table. Time after dropping (s) Position (m) 0 0 1 -1 2 -2 3 -5 4 -7 5 -14 172. [T] The Holling type II equation is described by f(x)=axn+x , where x is the amount of prey available and a >0 is the maximum consumption rate of the predator. a. Graph the Holling type II equation given a = 0.5 and n = 5. What are the differences between the Holling type I and II equations? b. Take the first derivative of the Holling type II equation and interpret the physical meaning of the derivative. c. Show that f(n)=12a and interpret the meaning of the parameter n. d. Find and interpret the meaning of the second derivative. What makes the Holling type II function more realistic than the Holling type I function?For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table. Time after dropping (s) Position (m) 0 0 1 -1 2 -2 3 -5 4 -7 5 -14 173. [T] The Holling type III equation is described by f(x)=ax2n2+x2 , where x is the amount of prey available and a >0 is the maximum consumption rate of the predator. Graph the Holling type III equation given a = 0.5 and n = 5. What are the differences between the Holling type II and III equations? Take the first derivative of the Holling type III equation and interpret the physical meaning of the derivative. Find and interpret the meaning of the second derivative (it may help to graph the second derivative). What additional ecological phenomena does the Holling type III function describe compared with the Holling type II function?[T] The populations of the snowshoe hare (in thousands) and the lynx (in hundreds) collected over 7 years from 1937 to 1943 are shown in the following table. The snowshoe hare is the primary prey of the lynx. Population of snowshoe hare (thousands) Population of lynx (hundreds) 20 10 55 15 65 55 95 60 Table 3.6 Snowshoe Hare and Lynx Populations Source: http://www.biotopics.co.uk/ newgcse/predatorpiey.html. Graph the data points and determine which Holling-type function fits the data best, Using the meanings of the parameters a and n, determine values for those parameters by examining a graph of the data. Recall that n measures what prey value results in the halfmaximum of the predator value. Plot the resulting Holling-type I, II, and III functions on top of the data. Was the result from part a. correct?For the following exercises, find dydx for the given functions. 175. y=x2secx+1 For the following exercises, find dydx for the given functions. 176. y=3cscx+5x For the following exercises, find dydx for the given functions. 177. y=x2cotx For the following exercises, find dydx for the given functions. 178. y=xx3sinx For the following exercises, find dydx for the given functions. 179. y=secxx For the following exercises, find dydx for the given functions. 180. y=sinxtanx For the following exercises, find dydx for the given functions. 181. y=(x+cosx)(1sinx)For the following exercises, find dydx for the given functions. 182. y=tanx1secx For the following exercises, find dydx for the given functions. 183. y=1cotx1+cotx For the following exercises, find dydx for the given functions. 184. y=cosx(1+cscx)For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 185. [T] f(x)=sinx,x=0 For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 186. [T] f(x)=cscx,x=2 For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 187. [T] f(x)=1+cosx,x=32 For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 188. [T] f(x)=secx,x=4 For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 189. [T] f(x)=x2tanxx=0 ]For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 190. [T] f(x)=5cotxx=4 For the following exercises, find d2ydx2 for the given functions. 191. y=xsinxcosx For the following exercises, find d2ydx2 for the given functions. 192. y=sinxcosx For the following exercises, find d2ydx2 for the given functions. 193. y=x12sinx For the following exercises, find d2ydx2 for the given functions. 194. y=1x+tanx For the following exercises, find d2ydx2 for the given functions. 195. y=2cscx For the following exercises, find d2ydx2 for the given functions. 196. y=sec2x Find all x values on the graph of f(x)=3sinxcosx where the tangent line is horizontal.Find all x values on the graph of f(x)=x2cosx for 0x2 where the tangent line has slope 2.Let f(x)=cotx . Determine the points on the graph of f for 0x2In where the tangent line(s) is (are) parallel to the line y=2x .[T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s(t)=6cost where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at t = 5 s.Let the position of a swinging pendulum in simple harmonic motion be given by s(t)=acost+bsint . Find the constants a and b such that when the velocity is 3 cm/s, s = 0 and t = 0.After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t)=5cost cm at t seconds after the jump. Sketch one period of the position function for t0 . Find the velocity function. Sketch one period of the velocity function for t0 . Determine the times when the velocity is 0 over one period. Find the acceleration function. Sketch one period of the acceleration function for t0 .The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y=10+5sinx where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y' and determine the intervals where the number of burgers being sold is increasing.[T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y=0.5+0.3cost , where t is months since January. Find y’ and use a calculator to determine the intervals where the amount of rain falling is decreasing. For the following exercises, use the quotient rule to derive the given equations.For the following exercises, use the quotient rule to derive the given equations. 205. ddx(cotx)=csc2x For the following exercises, use the quotient rule to derive the given equations. 206. ddx(secx)=secxtanx For the following exercises, use the quotient rule to derive the given equations. 207. ddx(cscx)=cscxcotx Use the definition of derivative and the identity cos(x+h)=cosxcoshsinxsinh to prove that d(cosx)dx=sinx .For the following exercises, find the requested higher-order derivative for the given functions. 209. d3ydx3ofy=3cosx For the following exercises, find the requested higher-order derivative for the given functions. 210. d2ydx2ofy=3sinx+x2cosx For the following exercises, find the requested higher-ordei derivative for the given functions. 211. d4ydx4ofy=5cosx For the following exercises, find the requested higher-order derivative for the given functions. 212. d2ydx2ofy=secx+cotx For the following exercises, find the requested higher-order derivative for the given functions. 213. d3ydx3ofy=x10secx For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 214. y=3u6,u=2x2 For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 215. y=6u3,u=7x4 For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 216. y=sinu,u=5x1 For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 217. y=cosu,u=x8 For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 218. y=tanu,u=9x+2 For the following exercises, given y=f(u) and u=g(x) , find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx . 219. y=4u+3,u=x26x For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 220. y=(3x2)6 For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 221. y=(3x2+1)3 For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 222. y=sin5(x)For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 223. y=(x7+7x)7 For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 224. y=tan(secx)For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 225. y=csc(x+1)For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 226. y=cot2x For each of the following exercises, decompose each function in the form y=f(u) and u=g(x), and find dydxas a function of x. 227. y=6sin3x For the following exercises, find dydx for each function. 228. y=(3x2+3x1)4 For the following exercises, find dydx for each function. 229. y=(52x)2 For the following exercises, find dydx for each function. 230. y=cos3(x)For the following exercises, find dydx for each function. 231. y=(2x3x2+6x+1)3 For the following exercises, find dydx for each function. 232. y=1sin2(x)For the following exercises, find dydx for each function. 233. y=(tanx+sinx)3 For the following exercises, find dydx for each function. 234. y=x2cos4x For the following exercises, find dydx for each function. 235. y=sin(cos7x)For the following exercises, find dydx for each function. 236. y=6+secx2 For the following exercises, find dydx for each function. 237. y=cot3(4x+1)Let y=[f(x)]3 and suppose that f(1)=4 and dydx=10 for x=1 . Find f(1) .Let y=(f(x)+5x2)4 and suppose that f(1)=4 and dydx=3 when x=1 . Find f(1)Let y=(f(u)+3x)2 and u=x32x . If f(4)=6 and dydx=18 when x=2 , find f(4) .[T] Find the equation of the tangent line of y=sin(x2) at the origin. Use a calculator to graph the function and the tangent line together.[T] Fine the equation of the tangent line to y=(3x+1x)2 at the point (1, 16). Use a calculator to graph the function and the tangent line together.Find the x-coordinates at which the tangent line to y=(x+6x)8 is horizontal.[T] Find an equation of the line that is normal to g()=sin2() . the point (14,12) . Use a calculator to graph the function and the normal line together,For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 245. h(x)=f(g(x));a=0 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 246. h(x)=g(f(x));a=0 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 247. h(x)=(x4+g(x))2;a=1 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 248. h(x)=( f( x ) g( x ))2;a=3 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 249. h(x)=f(x+f(x));a=1 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 250. h(x)=(1+g(x))3;a=2 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 251. h(x)=g(2+f(x2));a=1 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 252. h(x)=f(g(sinx));a=0 For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 253. [T] The position function of a freight train is given by s(t)=100(t+1)2 , with s in meters and t in seconds. At time t=6s, find the trains For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 254. [T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function. Where t is measured in seconds and s is in inches: s(t)=3cos(t+4) . a. Determine the position of the spring at t = 1.5 s b. Find the velocity of the spring at t = 1.5 s.For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 255. [T] The total cost to produce x boxes of Thin mint Girl Scout cookies is C dollars, where C=0.0001x30.02x2+3x+300 . In t weeks production is estimated to be x=1600+100t boxes. a. Find the marginal cost C(x) . b. Use Leibniz’s notation for the chain rule, dCdt=dCdxdxdt , to find the rate with respects to time t that the cost is changing. c. Use b. to determine how fast costs are increasing when t = 2 weeks. Include units with the answer.For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 256. [T] The formula for the area of a circle is A=r2 , where r is the radius of the circle. Suppose a circle is expanding, meaning that both the area A and the radius r (in inches) are expanding. a. Suppose r=2100( t+7)2 where t is time in seconds. Use the chain rule dAdt=dAdrdrdt to find the rate at which the area is expanding. b. Use a. to find the rate at which the area is expanding t = 4 s For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 257. [T] The formula for the volume of a sphere is S=43r3 , where r (in feet) is the radius of the sphere. Suppose a spherical snowball in melting in the sun. a. Suppose r=1( t+1)2112 where t is time in minutes. Use the chain rule dSst=dSdrdrdt to find the rate at which the snowball is melting. b. Use a. to find the rate at which the volume is changing at t = 1 min.For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 258. [T] The daily temperature in degree Fahrenheit of Phoenix in the summer can be modeled by the function T(x)=9410cos[12(x2)] , where x is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.For the following exercises, use the information in the following table to find h(a) at the given value for a. x f(x) f(x) g(x) g(x) 0 2 5 0 2 1 1 -2 3 0 2 4 4 1 -1 3 3 -3 2 3 259. [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D(t)=5sin(6t76)+8 , where t is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.For the following exercises, use the graph of y=f(x) to sketch the graph of y=f1(x), and use part a. to estimate (f1)(1) . 260.For the following exercises, use the graph of y=f(x) to sketch the graph of y=f1(x), and use part a. to estimate (f1)(1) . 261.For the following exercises, use the graph of y=f(x) to a. sketch the graph of y=f1(x), and b. use part a. to estimate (f1)(1).For the following exercises, use the graph of y=f(x) to sketch the graph of y=f1(x), and use part a. to estimate (f1)(1) . 263.For the following exercises, use the functions y=f(x) to find dfdxatx=a and x=f1(y) Then use part b. to find df1dy at y=f(a) . 264. f(x)=6x1,x=2 For the following exercises, use the functions y=f(x) to find dfdxatx=a and x=f1(y) Then use part b. to find df1dy at y=f(a) . 265. f(x)=2x33,x=1 For the following exercises, use the functions y=f(x) to find dfdxatx=a and x=f1(y) Then use part b. to find df1dy at y=f(a) . 266. f(x)=9x2,0x3,x=2 For the following exercises, use the functions y=f(x) to find dfdxatx=a and x=f1(y) Then use part b. to find df1dy at y=f(a) . 267. f(x)=sinx,x=0 For each of the following functions, find (f1)(a) . 268. f(x)=x2+3x+2,x1,a=2 For each of the following functions, find (f1)(a) . 269. f(x)=x3+2x+3,a=0 For each of the following functions, find (f1)(a) . 270. f(x)=x+x,a=2 For each of the following functions, find (f1)(a) . 271. f(x)=x2x,x0,a=1 For each of the following functions, find (f1)(a) . 272. f(x)=x+sinx,a=0 For each of the following functions, find (f1)(a) . 273. f(x)=tanx+3x2,a=0 For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 274. f(x)=41+x2,p(2,1)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 275. f(x)=x4,p(2,8)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 276. f(x)=(x3+1)4,p(16,1)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 277. f(x)=x3x+2,p(8,2)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 278. f(x)=x5+3x34x8,p(8,1)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 279. y=sin1(x2)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 280. y=cos1(x)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1 at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 281. y=sec1(1x)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 282. y=csc1x For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 283. y=(1+ tan 1x)3 For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 284. y=cos1(2x)sin1(2x)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 285. y=1tan1(x)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 286. y=sec1(x)For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 287. y=cot14x2 For each of the given function y=f(x) . Find the slope of the tangent line to its inverse function f1at the indicated point P, and Find the equation of the tangent line to the graph of f1 at the indicated point. 288. y=xcsc1x For the following exercises, use the given values to find (f1)(a) . 289. f()=0,f()=1,a=0 For the following exercises, use the given values to find (f1)(a) . 290. f(6)=2,f(6)=13,a=2 For the following exercises, use the given values to find (f1)(a) . 291. f(13)=8,f(13)=2,a=8 For the following exercises, use the given values to find (f1)(a) . 292. f(3)=12,f(3)=23,a=12 For the following exercises, use the given values to find (f1)(a) . 293. f(1)=3,f(1)=10,a=3 For the following exercises, use the given values to find (f1)(a) . 294. f(1)=0,f(1)=2,a=0 [T] The position of a moving hockey puck after t seconds is s(t)=tan1t where s is in meters. Find the velocity of the hockey puck at any time t. Find the acceleration of the puck at any time t. Evaluate a. and b. for t= 2, 4, and 6 seconds. What conclusion can be drawn from the results in c.?[T] A building that is 225 feet tall casts a shadowof various lengths x as the day goes by. An angle of elevation 0 is formed by lines from die top and bottom of the building to the tip of the shadow, as seen in the following figure. Find the rate of change of the angle ofelevation ddx when x = 272 feet.[T] A pole stands 75 feet tall. An angle is formed when wires of various lengths of x feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle ddx when a wire of length 90 feet is attached.[T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by =tan1(x2000) where x is the height of the rocket.Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.[T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle (in radians) given by =cot1x40cot1x10 . where x is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure. Find ddx . Evaluate ddx for x=5, 10, 15 and 20. Interpret the result in b.. Evaluate ddx for x = 25, 30, 35 and 40 Interpret the result in d. At what distance x should the person stand to maximize his or her viewing angle?For the following exercises, use implicit differentiation to find dydx . 300. x2y2=4 For the following exercises, use implicit differentiation to find dydx . 301. 6x2+3y2=12 For the following exercises, use implicit differentiation to find dydx . 302. x2y=y7 For the following exercises, use implicit differentiation to find dydx . 303. 3x3+9xy2=5x3 For the following exercises, use implicit differentiation to find dydx . 304. xycos(xy)=1 For the following exercises, use implicit differentiation to find dydx . 305. yx+4=xy+8 For the following exercises, use implicit differentiation to find dydx . 306. xy2=x7 For the following exercises, use implicit differentiation to find dydx . 307. ysin(xy)=y2+2 For the following exercises, use implicit differentiation to find dydx . 308. (xy)2+3x=y2 For the following exercises, use implicit differentiation to find dydx . 309. x3y+xy3=8 For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 310. [T] x4yxy3=2,(1,1)For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 311. [T] x2y2+5xy=14,(2,1)For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 312. [T] tan(xy)=y,(4,1)For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 313. [T] xy2+sin(y)2x2=10,(2,3)For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 314. [T] xy+5x7=34y,(1,2)

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