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All Textbook Solutions for Calculus
99E100E101E102E103E104EUsing Relationships In Exercises 105-108, use the given information to find f (2). g(2)=3andg'(2)=2 f(x)=g(x)h(x)106E107ESketching a Graph Sketch the graph of a differentiable function f such that f0 and f0 for all real numbers r Explain how you found your answer.109EIdentifying Graphs In Exercises 111 and 112, the graphs of f, f, and f are shown on the same set of coordinate axes. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to Math Grapht.com.111E112E113E114E115E116E117EHOW DO YOU SEE IT? The figure shows the graphs of the position, velocity, and acceleration functions of a particle. (a) Copy the graphs of the functions shown. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com. (b) On your sketch, identify when the panicle speeds up and when it slows down. Explain your reasoning.119EFinding a Pattern In Exercises 121 and 122. develop a general rule for f(n)(x) given f ( x ). f(x)=1x121EFinding a Pattern Develop a general rule for the nth derivative of xf (x), where f is a differentiable function of x.123E124E125E126E127E128E129E130E131E132E133E134E135E137E136EDecomposition of a Composite Function In Exercises 16, complete the table. y=f(g(x))u=g(x)y=f(u) y=(5x8)4 ______ ______Decomposition of a Composite Function In Exercises 16, complete the table. y=f(g(x))u=g(x)y=f(u) y=1x+1 ______ ______3E4E5E6E7E8E9E10E11E12E13EFinding a Derivative In Exercises 734, find the derivative of the function. f(x)=x24x+215E16E17E18E19E20E21E22E23E24E25E26E27E28E29EFinding a Derivative In Exercises 9-34, find the derivative of the function. h(t)=(t2t3+2)231E32E33E34E35EFinding a Derivative Using Technology In Exercises 55-60, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. y=2xx+137E38E39E40ESlope of a Tangent Line In Exercises 41 and 42, find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval [0, 2]. What can you conclude about the slope of the sine function sin ax at the origin?42E43E44E45E46EFinding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. y=sin(x)2Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. y=csc(12x)2Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. h(x)=sin2xcos2x50E51E52E53EFinding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. g(t)=5cos2t55E56EFinding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. f()=14sin2258E59E60E61E62E63EFinding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. y=cossin(tanx)65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88EFinding a Second Derivative In Exercises 83-88, find the second derivative of the function. f(x)=sinx290E91E92E93E94E95EIdentifying Graphs In Exercises 93 and 94, the graphs or a function f and its derivative f are shown. Label the graphs as f or f and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to MathGraphs.com.WRITING ABOUT CONCEPTS Identifying Graphs In Exercises 9598, the graphs of a function f and its derivative f' are shown. Label the graphs as f or f' and write a short paragraph stating the criteria you used in making your selection. To print an enlarged copy of the graph, go to Math Graphs.com.98EDescribing a Relationship In Exercises 99 and 100, the relationship between f and g is given. Explain the relationship between f' and g'. g(x)=f(x2)Think about It The table shows some values of the derivative of an unknown function f. Complete the table by finding the derivative of each transformation of f, if possible. (a) g(x)=f(x)2 (b) h(x)=2f(x) (c) r(x)=f(3x) (d) s(x)=f(x+2) x 2 1 0 1 2 3 f(x) 4 23 13 1 2 4 g(x) h(x) r(x) s(x)102EFinding Derivatives In Exercises 99 and 100, the graphs of f and g are shown. Let h(x)=f(g(x)) and s(x)=g(f(x)) . Find each derivative, if it exists. If the derivative does not exist, explain why. (a) Find h(1), (b) Find s'(5).104EDoppler Effect The frequency F of a fire truck siren heard by a stationary observer is F=132,400331v where v represents the velocity of the accelerating fire truck in meters per second (see figure). Find the rate of change of F with respect to v when (a) the fire truck is approaching at a velocity of 30 meters per second (use v ). (b) the fire truck is moving away at a velocity of 30 meters per second (use +v ). F=132,400331+v F=132,400331v106E107EWave Motion A buoy oscillates in simple harmonic motion y=Acost as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its highpoint. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t=0 (b) Determine the velocity of the buoy as a function of t109E110E111E112E113EConjecture Let f be a differentiable function of period p. (a) Is the function f periodic? Verify your answer. (b) Consider the function g(x)=f(2x). Is the function g'(x) periodic? Verify your answer.115E116EEven and Odd Functions (a) Show that the derivative of an odd function is even. That is, if f(x)=f(x), then f(x)=f(x). (b) Show that the derivative of an even function is odd. That is, if f(x)=f(x), then f(x)=f(x).118E119E120E121E122E123E124E125E126E127E128E129E130EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. x2+y2=92EFinding a Derivative In Exercises 116, find dy/dx by implicit differentiation. x1/2+y1/2=164EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. x3xy+y2=76EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. x3y3y=xFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. xy=x2y+1Finding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. x33x2y+2xy2=1210EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. sinx+2cos2y=112EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. cscx=x(1+tany)14EFinding a Derivative In Exercises 5-20, find dy / dx by implicit differentiation. y=sinxy16EFinding Derivatives Implicitly and Explicitly In Exercises 21-24. (a) Find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx implicitly and show that the result is equivalent to that of part (c). x2+y2=6418E19E20EFinding the Slope of a Graph In Exercises 25-32, find dy / dx by implicit differentiation. Then find the slope of the graph at the given point. xy=6,(6,1)Finding and Evaluating a Derivative In Exercises 2128, find dy/dx by implicit differentiation and evaluate the derivative at the given point. y3 x2 = 4, (2, 2)23EFinding and Evaluating a Derivative In Exercises 2128, find dy/dx by implicit differentiation and evaluate the derivative at the given point. x2/3+y2/3=5,(8,1)Finding the Slope of a Graph In Exercises 25-32, find dy / dx by implicit differentiation. Then find the slope of the graph at the given point. (x+y)3=x3+y3,(1,1)26E27E28E29E30EFamous Curves In Exercises 39-42, find the slope of the tangent line to the graph at the given point. Bifolium; (x2+y2)2=4x2y32E33E34EFamous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to MathGraphs.com.
35. Rotated hyperbola
Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to MathGraphs.com.
36. Rotated ellipse
37E38E39E40EEllipse (a) Use implicit differentiation to find an equation of the tangent line to the ellipse x22+y28=1 at (1,2). (b) Show that the equation of the tangent line to the ellipse x2a2+y2b2=1 at (x0,y0) is x0xa2+y0yb2=1.42E43E44EFinding a Second Derivative In Exercises 49-54. find d2y/dx2 implicitly in terms of x and y. x2+y2=446E47E48EFinding a Second Derivative In Exercises 4550, find d2y/dx2 implicitly in terms of x and y. y2 = x350EFinding an Equation of a Tangent Line In Exercises 55 and 56, use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window. x+y=5,(9,4)52ETangent Lines and Normal Lines In Exercises 63 and 64, find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the circle, the tangent lines, and the normal lines. x2+y2=25(4,3),(3,4)Tangent Lines and Normal Lines In Exercises 57 and 58, find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point. I Use a graphing utility to graph the circle, the tangent lines, and the normal lines. x2+y2=36(6,0),(5,11)55E56E57E58E59E60E61E62E63E64E65E66EOrthogonal Trajectories The figure below shows the topographic map carried by a group of hikers. The hikers are in a wooded area on top of the hill shown on the map, and they decide to follow the path of steepest descent (orthogonal trajectories to the contours on the map). Draw their routes if they start from point A and if they start from point H. Their goal is to reach the road along the top of the map. Which starting point should they use? To print an enlarged copy of the map, go to MathGraphs.com.68E69E70E71E72E73ENormals to a Parabola The graph shows die normal lines from the point (2.0) to the graph of the parabola x=y2. How many normal lines are there from the point (xn,0) to die graph of the parabola if (a) x0=14,. and (c) x0=12, (d) For what value of x0 are two of die normal lines perpendicular to each other?75ERelated Rates In your own words, state the guidelines for solving related-rate problems.Using Related Rates In Exercises 3-6, assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. Equation Find Given y=x (a) dydtwhenx=4 dxdt=3 (b) dxdtwhenx=25 dydt=2Using Related Rates In Exercises 3-6, assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. Equation Find Given y=3x25x (a) dydtwhenx=3 dxdt=2 (b) dxdtwhenx=2 dydt=43EUsing Related Rates In Exercises 3-6, assume that r and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. Equation Find Given x2+y2=25 (a) dydtwhenx=3,y=4 dxdt=8 (b) dxdtwhenx=4,y=3 dydt=-2Moving Point In Exercises 7-10, a point is moving along the graph of the given function at the rate dx/dt . Find dy/dt for the given values of x. y=2x2+1;dxdt=2 centimeters per second (a) x=1 (b) x=0 (c) x=1Moving Point In Exercises 7-10, a point is moving along the graph of the given function at the rate dx/dt . Find dy/dt for the given values of x. y=11+x2;dxdt=6 inches per second (a) x=2 (b) x=0 (c) x=2Moving Point In Exercises 710, a point is moving along the graph of the given function at the rate dx/dt.Find dy/dt for the given values of x. y=tanx;dxdt=3 feet per second. (a) x=3 (b) x=4 (c) x=0Moving Point In Exercises 7-10, a point is moving along the graph of the given function at the rate dx/dt . Find dy/dt for the given values of x. y=cosx;dxdt=4 centimeters per second (a) x=6 (b) x=4 (c) x=39EArea The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when (a) r = 8 centimeters and (b) r = 32 centimeters.12. Area The included angle of the two sides of constant equal length s of an isosceles triangle is .
(a) Show that the area of the triangle is given by
(b) The angle is increasing at the rate of radian per minute. Find the rates of change of the area when = π/6 and = π/3.
(c) Explain why the rate of change of the area of the triangle is not constant even though d/dt is constant.
13E14E15E16EHeight At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? (Hint: The formula for the volume of a cone is V=13r2h.)Height The volume of oil in a cylindrical container is increasing at a rate of 150 cubic inches per second. The height of the cylinder is approximately ten times the radius. At what rate is the height of the oil changing when the oil is 35 inches high? (Hint: The formula for the volume of a cylinder is V=r2h.)Depth A swimming pool is 12 meters long, 6 meters wide. 1 meter deep at the shallow end, and 3 meters deep at the deep end (see figure). Water is being pumped into the pool at 14 cubic meter per minute, and there is 1 meter of water at the deep end. (a) What percent of the pool is filled? (b) At what rate is the water level rising?Depth A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet (a) Water is being pumped into the trough at 2 cubic feet per minute. How fast is the water level rising when the depth h is 1 foot? (b) The water is rising at a rate of 38 inch per minute when h=2 feet. Determine the rate at which water is being pumped into the trough.Moving Ladder A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.Construction A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?Construction A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of 0.2 meter per second. Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y=6 meters.Boating A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see Figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?Air Traffic Control An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other (see figure). One plane is 225 miles from the point, moving at 450 miles per hour. The other plane is 300 miles from the point, moving at 600 miles per hour. (a) At what rate is the distance s between the planes decreasing? (b) How much time does the air traffic controller have to get one of the planes on a different flight path?Air Traffic Control An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure). When the plane is 10 miles away (s=10), the radar detects that the distance 5 is changing at a rate of 240 miles per hour. What is the speed of the plane?Sports A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 25 feet per second is 20 feet from third base. At what rate is the players distance from home plate changing?28EShadow Length A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). (a) When he is 10 feet from the base of the light, at what rate is the tip of his shadow moving? (b) When he is 10 feet from the base of the light, at what rate is the length of his shadow changing?Shadow Length Repeal Exercise 29 for a man 6 feet tall walking at a rate of 5 feet per second toward a light that is 20 feet above the ground (see figure).Machine Design The endpoints of a movable rod of length 1 meter have coordinates (x, 0) and (0, y) (see figure). The position of the end on the x-axis is x(t)=12sint6 where t is the time in seconds. (a) Find the time of one complete cycle of the rod. (b) What is the lowest point reached by the end of the rod on the y-axis? (c) Find the speed of the y-axis endpoint when the x-axis endpoint is (14,0)32E33EHOW DO YOU SEE IT? Using the graph of f, (a) determine whether dy/dt is positive or negative given that dx/dt is negative, and (b) determine whether dx/dt is positive or negative given that dy/dt is positive. Explain35EAdiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the equation pV1.3 = k, where k is a constant. Find the relationship between the related rates dp/dt and dV/dt.37E38EAngle of Elevation A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line from the end of the rod to the water?Angle of Elevation An airplane flies at an altitude of 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation is changing when the angle is (a). =30, (b). =60, and (c) =75.Linear vs. Angular Speed A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes angles of (a) =30, (b) =60 and (c) =70 with the perpendicular line from the light to the wall?Linear vs. Angular Speed A wheel of radius 30 centimeters revolves at a rate of 10 revolutions per second. A dot is painted at a point P on the rim of the wheel (see figure). (a) Find dx/dt as a function of . (b) Use a graphing utility to graph the function in part (a). (c) When is the absolute value of the rate of change of % greatest? When is it least? (d) Find dx/dt when =30 and =60.Flight Control An airplane is flying in still air with an airspeed of 275 miles per hour. The plane is climbing at an angle of 18. Find the rate at which it is gaining altitude.Security Camera A security camera is centered 50 feet above a 100-foot hallway (see figure). It is easiest to design the camera with a constant angular rate of rotation, but this results in recording the images of the surveillance area at a variable rate. So, it is desirable to design a system with a variable rate of rotation and a constant rate of movement of the scanning beam along the hallway. Find a model for the variable rate of rotation when | dx/dt |=2 feet per second.45EAcceleration In Exercises 49 and 50, find the acceleration of the specified object. (Hint; Recall that if a variable is changing at a constant rate, then its acceleration is zero.) Find the acceleration of the top of the ladder described in Exercise 21 when the base of the ladder is 7 feet from the wall.Acceleration In Exercises 49 and 50, find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, then its acceleration is zero.) Find the acceleration of the boat in Exercise 24(a) when there is a total of 13 feet of rope out.48EMoving Shadow A ball is dropped from a height of 20 meters. 12 meters away from the top of a 20-meter lamppost (see figure), The ball's shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released? (Submitted by Dennis Gittinger, St. Philips College, San Antonio. TX)Finding the Derivative by the Limit Process In Exercises 1-4, find the derivative of the function by the limit process. f(x)=122RE3RE4RE5RE6RE7RE8RE