Students! (1) with calculator Let R be the region bounded by the graphs of y = sin(π x) and y = x 3 4x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line y = splits the region R into two parts. Write, but do not evaluate, an integral expression for the area of the part of R that is below this horizontal line. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid. (d) The region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h(x) = 3 x. Find the volume of water in the pond. () No calculator Let R be the region in the first quadrant bounded by the graph of y = x, the horizontal line y = 6, and the y- axis, as shown in the figure above. (a) Find the area of r. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 7. (c) Region R is the base of a solid. For each y, where y 6, the cross section for the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid. 4
(3) No calculator Let R be the shaded region bounded by the graph of y = xe x, the line y = x, and the vertical line x = 1, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y =. (c) Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of R. (4) with calculator ( ) x(t) = t 4t + 8 A particle is moving along a curve so that its position at times t is x(t), y(t), where and y(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that dy. dt = tet 3 1 (a) Find the speed of the particle at t = 3 seconds. (b) Find the total distance traveled by the particle for t < 4 seconds. (c) Find the time t, t 4, when the line tangent to the path of the particle is horizontal. Is the direction of the motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of t when this occurs (ii) The slopes of the lines tangent to the particle's path at that point (iii) The y-coordinate of that point, given y() = 3+ 1 e (5) with calculator At time t, a particle moving in the xy-plane is at position (x(t), y(t)), where x(t) and y(t) are not explicitly given. dx dy For t, = 4t +1 and. At time t =, x() = and y() = 4. dt dt = sin ( t ) (a) Find the speed of the particle at time t = 3, and find the acceleration vector of the particle at time t = 3. (b) Find the slope of the line tangent to the path of the particle at time t = 3. (c) Find the position of the particle at time t = 3. (d) Find the total distance traveled by the particle over the time interval t 3. 5
(6) with calculator For t, a particle is moving along a curve so that its position at time t is ( x(t), y(t) ). At time t =, the particle dx is at position (1, 5). It is known that and. dt = t + dy e t dt = sin t (a) Is the horizontal movement of the particle to the left or to the right at time t =? Explain your answer. Find the slope of the path of the particle at time t =. (b) Find the x-coordinate of the particle's position at time t = 4. (c) Find the speed of the particle at time t = 4. Find the acceleration vector of the particle at time t = 4. (d) Find the distance traveled by the particle from time t = to t = 4. (7) with calculator A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time t seconds after she leaps, the horizontal distance from the front edge of the platform to the diver s shoulders is given by x(t), the vertical distance from the water surface to her shoulders is given by y(t), where x(t) and y(t) are measured in meters. Suppose that the diver's shoulders are 11l4 meters above the water when she makes her leap and that for t A, where A is the time that the diver's shoulders enter the water. (a) Find the maximum vertical distance from the water surface to the diver's shoulders. (b) Find A, the time that the diver's shoulders enter the water. (c) Find the total distance traveled by the diver's shoulders from the time she leaps from the platform until the time her shoulders enter the water. (d) Find the angle θ, < θ < π, between the path of the diver and the water at the instant the diver's shoulders enter the water. dx dt =.8 and dy dt = 3.6 9.8t, 6
(8) No calculator A particle moves along the x-axis so that its velocity at time t, for t 6, is given by a differentiable function v whose graph is shown above. The velocity is at time t =, t = 3, and t = 5, and the graph has horizontal tangents at t = 1 and t = 4. The areas of the regions bounded by the t-axis and the graph of v on the intervals [, 3], [3, 5], and [5, 6] are 8, 3, and, respectively. At time t =, the particle is at x =. (a) For t 6, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer. (b) For how many values of t, where t 6, is the particle at x = 8? Explain your reasoning. (c) On the interval < t < 3, is the speed of the particle increasing or decreasing? Give a reason for your answer. (d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer. (9) with calculator t (hours) 1 3 4 7 8 9 L(t ) (people) 1 156 176 16 15 8 Concert tickets went on sale at noon (t = ) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for t 9. Values of L(t) at various times t are shown in the table above. (a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:3 P.M. (t = 5.5). Show the computations that lead to your answer. Indicate units of measure. (b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale. (c) For t < 9, what is the fewest number of times at which L'(t) must equal? Give a reason for your answer. (d) The rate at which tickets were sold for t 9 is modeled by r(t) = 55te t/ tickets per hour. Based on the model, how many tickets were sold by 3 P.M. (t = 3), to the nearest whole number? 7
(1) with calculator Caren rides her bicycle along a straight road from home to school, starting at home at time t = minutes and arriving at school at time t = 1 minutes. During the time interval t 1 minutes, her velocity v(t), in miles per minute, is modeled by the piecewise-linear function whose graph is shown above. (a) Find the acceleration of Caren s bicycle at time t = 7.5 minutes. Indicate units of measure. (b) Using correct units, explain the meaning of v(t) dt in terms of Caren's trip. Find the value of 1 v(t) dt. (c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. (d) Larry also rides his bicycle along a straight road from home to school in 1 minutes. His velocity is modeled by the function w given by w(t) = π, where w(t) is in miles per minute for t 1 15 sin π 1 t minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. (11) with calculator 1 The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R(t) = 138t 675t 3 for t hours; R(t) is measured in people per hour. No one is in the auditorium at time t =, when the doors open. The doors close and the concert begins at time t =. (a) How many people are in the auditorium when the concert begins? (b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. (c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by w (t) = ( t)r(t). Find w() w(1), the total wait time for those who enter the auditorium after time t = 1. (d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c). 1 BC 1 8
(1) with calculator There is no snow on Janet's driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by f (t) = 7te cost cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A.M. (t = 6). The rate g(t), in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by for t < 6 g(t) = 15 for 6 t < 7 18 for 7 t < 9. (a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.? (b) Find the rate of change of the volume of snow on the driveway at 8 A.M. (c) Let h(t) represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain t 9. (d) How many cubic feet of snow are on the driveway at 9 A.M.? (13) with calculator t (hours) E(t ) (hundreds of entries) 5 7 8 4 13 1 3 A zoo sponsored a one-day contest to name a baby elephant. Zoo visitors deposited entries in a special box between noon (t = ) and 8 P.M. (t = 8). The number of entries in the box t hours after noon is modeled by a differentiable function E for t 8. Values of E(t), in hundreds of entries, at various times t are shown in the table above. (a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time t = 6. Show the computations that lead to your answer. 1 8 (b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of. 8 E(t)dt 1 8 Using correct units, explain the meaning of in terms of the number of entries. 8 E(t)dt (c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where P(t) = t 3 3t + 98t 976 hundreds of entries per hour for 8 t 1. According to the model, how many entries had not yet been processed by midnight (t = 1)? (d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer. 9
(14) with calculator t (minutes) H(t ) (degrees Celsius) 5 9 1 66 6 5 44 43 As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t 1, where time t is measured in minutes and temperature H(t) is measured in degrees Celsius. Values of H(t) at selected values of time t are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer. 1 1 (b) Using correct units, explain the meaning of in the context of this problem. Use a trapezoidal 1 H(t)dt 1 1 sum with the four subintervals indicated by the table to estimate H(t)dt. 1 (c) Evaluate 1 H (t)dt. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time t =, biscuits with temperature 1 C were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which is known that B (t) = 13.84e.173t. Using the given models, at time t = 1, how much cooler are the biscuits than the tea? (15) with calculator t (minutes) 4 9 15 W (t ) (degrees Fahrenheit) 55. 57.1 61.8 67.9 71. The temperature of water in a tub at time t is modeled by a strictly increasing, twice differentiable function W, where W(t) is measured in degrees Fahrenheit and t is measured in minutes. At time t =, the temperature of the water is 55 F. The water is heated for 3 minutes, beginning at time t =. Values of W(t) at selected times t for the first minutes are given in the table above. (a) Use the date in the table to estimate W'(1). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (b) Use the date in the table to evaluate in the context of this problem. (c) For t, the average temperature of the water in the tub is W (t)dt. Using correct units, interpret the meaning of W (t)dt. Use a left Riemann sum 1 with the four subintervals indicated by the data in the table to approximate. Does this W (t)dt approximation overestimate or underestimate the average temperature of the water over these minutes? Explain your reasoning. (d) For t 5, the function W that models the water temperature has first derivative given by W (t) =.4 t cos(.6t). Based on the model, what is the temperature of the water at time t = 5? 1 W (t)dt 1
(16) with calculator On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by G(t) = 9 + 45cos t, where t is measured in hours and 18 t 8. At the beginning of the workday (t = ), the plant has 5 tons of unprocessed gravel. During the hours of operation t 8,the plant processes gravel at a constant rate of 1 tons per hour. (a) Find G (5). Using correct units, interpret your answer in the context of the problem. (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t = 5 hours? Show the work that leads to your answer. (d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer. (17) with calculator Grass clippings are placed in a bin, where they decompose. For t 3, the amount of grass clippings remaining in the bin is modeled by A(t) = 6.687(.931) t, where A(t) is measured in pounds and t is measured in days. (a) Find the average rate of change of A(t) over the interval t 3. Indicate units of measure. (b) Find the value of A'(15). Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings remaining in the bin over the interval t 3. (d) For t > 3, L(t), the linear approximation to A at t = 3, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. 11
(18) No calculator t (minutes) C(t ) (ounces) 1 3 4 5 6 5.3 8.8 11. 1.8 13.8 14.5 Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time t, t 6, is given by a differentiable function C, where t is measured in minutes. Selected values of C(t), measured in ounces, are given in the table above. (a) Use the data in the table to approximate C'(3.5). Show the computations that lead to your answer, and indicate units of measure. (b) Is there a time t, t 4, at which C'(t) =? Justify your answer. (c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate 6 the value of 1 C(t)dt 6. Using correct units, explain the meaning of 1 C(t)dt 6 in the context of the problem. (d) The amount of coffee in the cup, in ounces, is modeled by B(t) = 16 16e.4t. Using this model, find the rate at which the amount of coffee in the cup is changing when t = 5. (19) No calculator 6 t (minutes) v A (t) (meters/minute) 5 8 1 1 4 1 15 Train A runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function v A (t), where time t is measured in minutes. Selected values for v A (t) are given in the table above. (a) Find the average acceleration of the train A over the interval t 8. (b) Do the data in the table support the conclusion that train A's velocity is 1 meters per minute at some time t with 5 < t < 8? Give a reason for your answer. (c) At time t =, train A's position is 3 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time t = 1. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time t = 1. (d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by v B (t) = 5t + 6t + 5, and at time t = the train is 4 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train b is changing at t =. 1
() No calculator The derivative of a function f is given by f (x) = (x 3)e x for x >, and ƒ(1) = 7. (a) The function f has a critical point at x = 3. At this point, does ƒ have a relative minimum, a relative maximum, or neither? Justify your answer. (b) On what intervals, if any, is the graph of f both decreasing and concave up? Explain your reasoning. (c) Find ƒ(3). (1) No calculator x 3 5 8 13 f (x) 1 4 3 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval x 13. (a) Estimate ƒ' (4). Show the work that leads to your answer. 13 (b) Evaluate ( 3 5 f (x))dx. Show the work that leads to your answer. (c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate f (x)dx. Show the work that leads to your answer. (d) Suppose ƒ' (5) = 3 and ƒ'' (x) < for all x in the closed interval 5 x 8. Use the tangent to the graph of ƒ at 4 x = 5 to show that ƒ(7) 4. Use the secant line for the graph of ƒ on 5 x 8 to show that ƒ(7). 3 () No calculator 13 Let f (x) = e x. Let R be the region in the first quadrant bounded by the graph of ƒ, the coordinate axes, and the vertical line x = k, where k >. The region R is shown in the figure above. (a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of R in terms of k. (b) The region R is rotated about the x-axis to form a solid. Find the volume, V, of the solid in terms of k. dk (c) The volume V, found in part b, changes as k changes. If, determine when. dt = 1 dv 3 dt k = 1 13
(3) No calculator The continuous function ƒ is defined on the interval 4 x 3. The graph of ƒ consists of two quarter circles and one line segment, as shown in the figure above. Let g(x) = x + f (t)dt. (a) Find g( 3). Find g'(x) and evaluate g'( 3). (b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval 4 x 3. Justify your answer. (c) Find all values of x on the interval 4 < x < 3 for which the graph of g has a point of inflection. Give a reason for your answer. (d) Find the average rate of change of ƒ on the interval 4 x 3. There is no point c, 4 < c < 3, for which f (c) is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. x (4) No calculator 14
The function ƒ is defined on the closed interval [ 5, 4]. The graph of ƒ consists of three line segments and is shown in the figure above. Let g be the function defined by g(x) = f (t)dt. (a) Find g(3). (b) On what open intervals contained in 5 < x < 4 is the graph of g both increasing and concave down? Give a reason for your answer. (c) The function h is defined by h(x) = g(x) 5x. Find h'(3). (d) The function p is defined by p(x) = f ( x x). Find the slope of the line tangent to the graph of p at the point where x = 1. x 3 (5) No calculator Let ƒ be the continuous function defined on [ 4, 3] whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let g be the function given by g(x) = f (t)dt. (a) Find the values of g() and g( ). (b) For each of g'( 3) and g''( 3), find the value or state that it does not exist. (c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (d) For 4 < x <3, find all values of x for which the graph of g has a point of inflection. Explain your reasoning. 1 x 15
(6) No calculator The figure above shows the graph of ƒ', the derivative of a twice-differentiable function ƒ, on the closed interval x 8. The graph of ƒ' has horizontal tangent lines at x = 1, x = 3, and x = 5. The areas of the regions between the graph of ƒ' and the x-axis are labeled in the figure. The function ƒ is defined for all real numbers and satisfies ƒ(8) = 4. (a) Find all values of x on the open interval < x < 8 for which the function ƒ has a local minimum. Justify your answer. (b) Determine the absolute minimum value of ƒ on the closed interval x 8. Justify your answer. (c) On what open intervals contained in < x < 8 is the graph of ƒ both concave down and increasing? Explain your reasoning. (d) The function g is defined by g(x) = ( f (x)) 3. If f (3) = 5, find the slope of the line tangent to the graph of g at x = 3. 16
(7) with calculator The graphs of the polar curves r = 3 and r = 4 sinθ are shown in the figure above. The curves intersect when θ = π 6 and θ = 5π 6. (a) Let S be the shaded region that is inside the graph of r = 3 and also inside the graph of r = 4 sinθ. Find the area of S. (b) A particle moves along the polar curve r = 4 sinθ so that at time t seconds, θ = t. Find the time t in the interval 1 t for which the x-coordinate of the particle's position is 1. (c) For the particle described in part b, find the position vector in terms of t. Find the velocity vector at time t = 1.5. (8) with calculator The graphs of the polar curves r = 3 and r = 3 sin(θ) are shown in the figure above for θ π. (a) Let R be the shaded region that is inside the graph of r = 3 and inside the graph of r = 3 sin(θ). Find the area of R. (b) For the curve r = 3 sin(θ), find the value of dx dθ at θ = π 6. (cont) 17
(c) The distance between the two curves changes for < θ < π. Find the rate at which the distance between the two curves is changing with respect to θ when θ = π 3. (d) A particle is moving along the curve r = 3 sin(θ) so that dθ = 3 for all times t. Find dt the value of dr at θ = π dt 6. (9) No calculator Consider the logistic differential equation dy dt = y (6 y). Let y = ƒ(t) be the particular solution to the 8 differential equation with ƒ() = 8. (a) A slope field for this differential equation is given below. Sketch possible solution curves through the points (3, ) and (, 8). (b) Use Euler s method, starting at t = with two steps of equal size, to approximate ƒ(1). (c) Write the second-degree Taylor polynomial for f about t =, and use it to approximate ƒ(1). (d) What is the range of ƒ for t? 18
(3) No calculator Consider the differential equation dy dx = 6x x y. Let y = ƒ(x) be a particular solution to this differential equation with initial condition ƒ( 1) =. (a) Use Euler s method with two steps of equal size, starting at x = 1, to approximate ƒ(). Show the work that leads to your answer. (b) At the point ( 1, ), the value of d y is 1. Find the second-degree Taylor polynomial for ƒ about x = 1. dx (c) Find the particular solution y = ƒ(x) to the given differential equation with the initial condition ƒ( 1) =. (31) No calculator dy Consider the differential equation. Let y = ƒ(x) be the particular solution to this differential dx = 1 y equation with the initial conditions ƒ(1) =. For this particular solution, ƒ(x) < 1 for all values of x. (a) Use Euler's Method, starting at x = 1 with two steps of equal size, to approximate ƒ(). Show the work that leads to your answer. (b) Find lim x 1. Show the work that leads to your answer. (c) Find the particular solution y = ƒ(x) to the differential equation ƒ(1) =. f (x) x 3 1 dy dx = 1 y with the initial condition (3) No calculator At the beginning of 1, a landfill contained 14 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential dw equation for the next years. W is measured in tons, and t is measured in years from dt the start of 1. (a) Use the line tangent to the graph of W at t = to approximate the amount of solid waste at the landfill contains at the end of the first 3 months of 1 (time t = 1 ). 4 d W d W (b) Find in terms of W. Use to determine whether your answer in part (a) is an underestimate or an dt dt overestimate of the amount of solid waste that the landfill contains at time t = 1. 4 (c) Find the particular solution W = W(t) to the differential equation W() = 14. = 1 ( 5 W 3 ) dw dt = 1 ( 5 W 3 ) with initial condition 19
(33) No calculator The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t =, when the bird is first weighted, its weight is grams. If B(t) is the weight of the bird, in grams, at time t days after it is first weighed, then db. dt = 1 (1 B) 5 Let y = B(t) be the solution to the differential equation above with initial condition B() =. (a) Is the bird gaining weight aster when it weights 4 grams or when it weights 7 grams? Explain your reasoning. d B d B (b) Find in terms of B. Use to explain why the graph of B cannot resemble the following graph. dt dt (c) Use separation of variables to find y = B(t), the particular solution to the differential equation with initial condition B() =. (34) No calculator Consider the differential equation dy dx = y (x + ). Let y = ƒ(x) be the particular solution to the differential equation with initial condition ƒ() = 1. f (x) +1 (a) Find lim. Show the work that leads to your answer. x sin x (b) Use Euler's method, starting at x = with two steps of equal size, to approximate f 1. (c) Find y = ƒ(x), the particular solution to the differential equation with initial condition ƒ() = 1.
(35) No calculator x 1 1.1 1. 1.3 1.4 f '(x) 8 1 1 13 14.5 The function ƒ is twice differentiable for x > with ƒ(1) = 15 and ƒ''(1) =. Values of ƒ', the derivative of ƒ, are given for selected values of x in the table above. (a) Write an equation for the line tangent to the graph of ƒ at x = 1. Use this line to approximate ƒ(1.4). (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate 1 1.4 f (x) dx. Use the approximation for f (x) dx to estimate the value of ƒ(1.4). Show the 1 1.4 computations that lead to your answer. (c) Use Euler's method, starting at x = 1 with two steps of equal size, to approximate ƒ(1.4). Show the computations that lead to your answer. (d) Write the second-degree Taylor polynomial for ƒ about x = 1. Use the Taylor polynomial to approximate ƒ(1.4). (36) No calculator The Taylor series for a function ƒ about x = 1 is given by n ( 1) n+1 (x 1)n and converges to n=1 n ƒ(x) for x 1 < R, where R is the radius of convergence of the Taylor series. (a) Find the value of R. (b) Find the first three nonzero terms and the general term of the Taylor series for ƒ', the derivative of ƒ, about x = 1. (c) The Taylor series for ƒ' about x = 1, found in part b, is a geometric series. Find the function ƒ' to which the series converges for x 1 < R. Use this function to determine ƒ for x 1 < R. 1
(37) with calculator x h(x) h' (x) h''(x) h'''(x) h (4) (x) 1 11 3 4 99 18 8 18 488 3 448 3 584 9 3 317 753 1383 4 3483 16 115 16 Let h be a function having derivatives of all orders for x >. Selected values of h and its first four derivatives are indicated in the table above. The function h and these four derivatives are increasing on the interval 1 x 3. (a) Write the first-degree Taylor polynomial for h about x = and use it to approximate h(1.9). Is this approximation greater than or less than h(1.9)? Explain your reasoning. (b) Write the third-degree Taylor polynomial for h about x = and use it to approximate h(1.9). (c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about x = approximates h(1.9) with error less than 3 x 1 4. (38) No calculator The Maclaurin series for e x is e x = 1+ x + x + x3 xn +...+ +... The continuous function ƒ is defined by 6 n! f (x) = x 1) e( 1 for x 1 and ƒ(1) = 1. The function ƒ has derivatives of all orders at x = 1. (x 1) (a) Write the first four nonzero terms and the general term of the Taylor series for e ( x 1) about x = 1. (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for ƒ about x = 1. (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for ƒ about x = 1 to determine whether the graph of ƒ has any points of inflection.
(39) No calculator cos x 1 for x f (x) = x 1 for x = The function ƒ, defined above, has derivatives of all orders. Let g be the function defined by x g(x) = 1+ f (t)dt. (a) Write the first three nonzero terms and the general term of the Taylor series for cos x about x =. Use this series to write the first three nonzero terms and the general term of the Taylor series for f about x =. (b) Use the Taylor series for f about x = found in part (a) to determine whether ƒ has a relative maximum, relative minimum, or neither at x =. Give a reason for your answer. (c) Write the fifth-degree Taylor polynomial for g about x =. (d) The Taylor series for g about x =, evaluated at x = 1, is an alternating series with individual terms that decrease in absolute value to. Use the third-degree Taylor polynomial for g about x = to estimate the value of g(1). Explain why this estimate differs from the actual value of g(1) by less than 1 6!. (4) No calculator Let f (x) = sin( x ) + cos x. The graph of y = f (5) (x) is shown above. (a) Write the first four nonzero terms of the Taylor series for sinx about x =, and write the first four nonzero terms of the Taylor series for sin( x ) about x =. (b) Write the first four nonzero terms for the Taylor series for cosx about x =. Use this series and the series for ( ), found in part (a), to write the first four nonzero terms of the Taylor series for ƒ about x =. sin x (c) Find the value of f (6) (). (d) Let P 4 (x) be the fourth-degree Taylor polynomial for ƒ about x =. Using information from the graph of 1 y = f (5) (x) shown above, show that P 4 4 f 1 4 < 1 3. 3
(41) No calculator The function g has derivatives of all orders, and the Maclaurin series for g is x n+1 ( 1) n = x n= n + 3 3 x3 5 + x5 7.... (a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g. (b) The Maclaurin series for g evaluated at x = 1 is an alternating series whose terms decrease in absolute value to. The approximation for g 1 17 using the first two nonzero terms of this series is. Show that this 1 approximation differs from g 1 by less than 1. (c) Write the first three nonzero terms and the general term of the Maclaurin series for g'(x). (4) No calculator A function ƒ has derivatives of all orders at x =. Let P n (x) denote the nth-degree Taylor polynomial for ƒ about x =. 1 (a) It is known that ƒ() = 4 and that P 1 = 3. Show that ƒ'() =. (b) It is known that f () = 3 and f () = 1 3. Find P (x). 3 (c) The function h has first derivative given by h'(x) = ƒ(x). It is known that h() = 7. Find the third-degree Taylor polynomial for h about x =. 4