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All Textbook Solutions for Calculus: An Applied Approach (MindTap Course List)
62E63E64E65EQuality Control The percent P of defective parts produced by a new employee t days after the employee starts work can be modeled by P=t+175050(t+2). Find the rate of change of P at (a) t = 1 and (b) t = 10.Environment The model P=t2-t+1t2+1 measures the percent P (in decimal form) of the normal level of oxygen in a pond, where t is the time (in weeks) after organic waste is dumped into the pond. Find the rate of change of Pat(a)t=0.5,(b)t=2,and(c)t=8. Interpret the meaning of these values.Physical Science The temperature T (in degrees Fahrenheit) of food placed in a refrigerator is modeled by T=10(4t2+16t+75t2+4t+10) where t is the time (in hours). What is the initial temperature of the food? Find the rate of change of T with respect to t at (a) t=1, (b) t=3, (c) t=5, and (d) t=10. Interpret the meaning of these values.69EHOW DO YOU SEE IT? The advertising manager for a new product determines that P percent of the potential market is aware of the product t weeks after the advertising campaign begins. (a) What happens to the percent of people who are aware of the product in the long run? (b) What happens to the rate of change of the percent of people who are aware of the product in the long run?71E72E73E74E75E76E77E1QY2QY3QY4QY5QY6QY7QY8QY9QY10QY11QY12QY13QY14QY15QY16QYIn Exercises 14-17, find the average rate of change of the function over the given interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. f(x)=x;3[8,27]The profit P (in dollars) from selling x units of a product is given by P=0.0125x2+16x600. (a) Find the marginal profit when x = 175. (b) Find the additional profit when sales increase from 175 to 176 units. (c) Compare the results of parts (a) and (b).19QY20QY21QY22QY23QYCheckpoint 1 Worked-out solution available at LarsonAppliedCalculus.com Write each function as the composition of two functions, where y=f(g(x)). a. y=1x+1 b. y=(x2+2x+5)32CPCheckpoint 3 Worked-out solution available at LarsonAppliedCalculus.com Find the derivative of y=(x2+3x)4.4CP5CP6CP7CP8CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWUIn Exercises 7-10, factor the expression. 4(x2+1)2x(x2+1)310SWUDecomposing Composite Functions In Exercises 1-6, identify the inside function, u=g(x), and the outside function, y=f(u). See Example 1. y=f(g(x))u=g(x)y=f(u)y=(6x5)42EDecomposing Composite Functions In Exercises 1-6, identify the inside function, u=g(x), and the outside function, y=f(u). See Example 1. y=f(g(x))u=g(x)y=f(u)y=5x2Decomposing Composite Functions In Exercises 1-6, identify the inside function, u=g(x), and the outside function, y=f(u). See Example 1. y=f(g(x))u=g(x)y=f(u)y=9x235E6EUsing the Chain Rule In Exercises 712, use the Chain Rule to find the derivative of the function. See Example 2. y=(4x+7)28E9EUsing the Chain Rule In Exercises 712, use the Chain Rule to find the derivative of the function. See Example 2. y=46x+54Using the Chain Rule In Exercises 7-12, use the Chain Rule to find the derivative of the function. See Example 2. y=(5x42x)2/312EChoosing a Differentiation Rule In Exercises 13-18, choose the rule that you would use to most efficiently find the derivative of the function. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule f(x)=21x314EChoosing a Differentiation Rule In Exercises 1318, choose the rule that you would use to most efficiently find the derivative of the function. Simple Power Rule Constant Rule General Power Rule Quotient Rule f(x)=82316EChoosing a Differentiation Rule In Exercises 1318, choose the rule that you would use to most efficiently find the derivative of the function. Simple Power Rule Constant Rule General Power Rule Quotient Rule f(x)=x2+9x3+4x26Choosing a Differentiation Rule In Exercises 1318, choose the rule that you would use to most efficiently find the derivative of the function. Simple Power Rule Constant Rule General Power Rule Quotient Rule f(x)=xx3+2x5Using the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. y=(2x7)320EUsing the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. h(x)=(6xx3)2Using the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. f(x)=(2x36x)4/323E24EUsing the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. s(t)=2t2+5t+226EUsing the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. f(x)=2(29x)328E29EUsing the general Power Rule In Exercises 1930, use the General Power Rule to find the derivative of the function. See Examples 3 and 5. y=1(4x3)4331E32EFinding an Equation of a Tangent Line In Exercises 3136, find an equation of the tangent line to the graph of f at the point (2, f(2)). Then use a graphing utility to graph the function and the tangent line in the same viewing window. See Example 4. f(x)=4x2734E35E36E37E38EUsing Technology In Exercises 37-40, use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. f(x)=x+1x40EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. y=14x242E43E44E45E46EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. y=1x+248EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. f(x)=x(3x9)350EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. y=x2x+352EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. y=t2t254EFinding Derivatives In Exercises 4156, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. y=(65xx21)256E57E58EFinding an Equation of a Tangent Line In Exercises 5764, find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. f(t)=36(3t)2;(0,4)60EFinding an Equation of a Tangent Line In Exercises 5764, find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. f(t)=(t29)t+2;(1,8)62EFinding an equation of a Tangent Line In Exercises 5764, find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. f(x)=x+12x3;(2,3)64EFinding Horizontal Tangent Lines In Exercises 6568, find the point(s), if any, at which the graph of f has a horizontal tangent line. f(x)=x2+4366EFinding Horizontal Tangent Lines In Exercises 6568, find the point(s), if any, at which the graph of f has a horizontal tangent line. f(x)=x2x168ECompound Interest You deposit $1000 in an account with an annual interest rate of r (in decimal form) compounded monthly. At the end of 5 years, the balance A is A=1000(1+r12)60. Find the rate of change of A with respect to r when (a) r = 0.08, (b) r = 0.10, and (c) r = 0.12.70E71EHOW DO YOU SEE IT? The cost C (In dollars) of producing x units of a product is C = 60x + 1350. For one week, management determined that the number of units produced x at the end of t hours was x=1.6t3+19t20.5t1. The graph shows the cost C in terms of the time t. Using the graph, which is greater, the rate of change of the cost after 1 hour or the rate of change of the cost after 4 hours? Explain why the cost function is not increasing at a constant rate during the eight-hour shift.73ECheckpoint 1 Worked-out solution available at LarsonAppliedCalculus.com Find the first four derivatives of f(x)=6x32x2+1.2CPCheckpoint 3 Worked-out solution available at LarsonAppliedCalculus.com Find the fourth derivative of y=1x24CP5CP6CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWU10SWUFinding Higher-Order Derivatives In Exercises 112, find the second derivative of the function. See Examples 1 and 3. f(x)=92x2E3E4EFinding Higher-Order Derivatives In Exercises 112, find the second derivative of the function. See Examples 1 and 3. g(t)=13t34t2+2t6E7E8EFinding Higher-Order Derivatives In Exercises 112, find the second derivative of the function. See Examples 1 and 3. f(x)=3(2x2)310E11E12E13E14EFinding Higher-Order Derivatives In Exercises 13-18, find the third derivative of the function. See Examples 1 and 3. f(x)=5x(x+4)316E17E18EFinding Higher-Order Derivatives In Exercises 1924, find the given value. See Example 2. FunctionValueg(t)=5t4+10t2+3g(2)20E21E22E23E24E25E26EFinding Higher-Order Derivatives In Exercises 2530, find the higher-order derivative. See Examples 1 and 3. GivenDerivativef(x)=4x4f(4)(x)28E29E30E31E32E33E34EVelocity and Acceleration A ball is propelled straight upward from ground level with an initial velocity of 144 feet per second. (a) Write the position, velocity, and acceleration functions of the ball. (b) Find the height, velocity, and acceleration at t = 3. (c) When is the ball at its highest point? How high is this point? (d) How fast is the ball traveling when it hits the ground? How is this speed related to the initial velocity?Velocity and Acceleration A brick becomes dislodged from the top of the Empire State Building (at a height of 1250 feet) and falls to the sidewalk below. (a) Write the position, velocity, and acceleration functions of the brick. (b) How long does it take the brick to hit the sidewalk? (c) How fast is the brick traveling when it hits the sidewalk?37EStopping Distance A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is given by s=8.25t2+66t, where s is measured in feet and t is measured in seconds. Use this function to complete the table showing the position, velocity, and acceleration for each given value of t. What can you conclude? t 0 1 2 3 4 s ds/dt d2s/dt239E40E41E42E43ETrue or False? In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The second derivative represents the rate of change of the first derivative.Project: Cell Phone Subscribers in U.S. For a project analyzing the number of cell phone subscribers in the United States from 2000 to 2013, visit this texts website at LarsonAppliedCalculus.com. (Source: CTIAThe Wireless Association)Checkpoint 3 Worked-out solution available at LarsonAppliedCalculus.com Find dy/dx for the equation x2y=1.2CP3CP4CP5CP6CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWU1EFinding Derivatives In Exercises 1-12, find dy/dx . See Examples 1 and 3. 2.3x2y=8xFinding Derivatives In Exercises 1-12, find dy/dx . See Examples 1 and 3. 3.y2=1x2,0x14EFinding Derivatives In Exercises 1-12, find dy/dx . See Examples 1 and 3. 5.y4y2+7y6x=96EFinding Derivatives In Exercises 1-12, find dy/dx . See Examples 1 and 3. 7.xy2+4xy=108EFinding Derivatives In Exercises 1-12, find dy/dx . See Examples 1 and 3. 9.2x+yx5y=110E11E12EFinding the Slope of a Graph Implicitly In Exercises 13-26, find the slope of the graph of the equation at the given point. See Examples 4 and 5. Equation Point 13.x2+y2=16 (0,4)14EFinding the Slope of a Graph Implicitly In Exercises 13-26, find the slope of the graph of the equation at the given point. See Examples 4 and 5. Equation Point 15.y+xy=4(5,1)16E17E18EFinding the Slope of a Graph Implicitly In Exercises 13-26, find the slope of the graph of the equation at the given point. See Examples 4 and 5. Equation Point 19.xyx=y (32,3)20EFinding the Slope of a Graph Implicitly In Exercises 13-26, find the slope of the graph of the equation at the given point. See Examples 4 and 5. Equation Point 21.x1/2+y1/2=9(16,25)22E23E24EFinding the Slope of a Graph Implicitly In Exercises 13-26, find the slope of the graph of the equation at the given point. See Examples 4 and 5. Equation Point 25.y2(x2+y2)=2x2(1,1)26E27E28E29E30E31EFinding the Slope of a Graph Implicitly In Exercises 27-32, find the slope of the graph at the given point. See Examples 4 and 5. (4-x)y2=x333E34E35E36E37E38E39E40E41E42EDemand In Exercises 43-46, find the rate of change of x with respect to p. See Example 6. 43.p=20.00001x3+0.1x,x044EDemand In Exercises 43-46, find the rate of change of x with respect to p. See Example 6. 45.p=200x2x,0x20046EProduction Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by 100x0.75y0.25=135,540. (a) Find the rate of change of y with respect to x when 1500 and y = 1000. (b) The model used in this problem is called the Cobb-Douglas production function. Graph the model on a graphing utility and describe the relationship between labor and capital.48E49ECheckpoint 1 Worked-out solution available at LarsonAppliedCalculus.com The variables x and y are differentiable functions of t and are related by the equation y=x3+2. When x=1,dx/dt, Find dy/dt when x = 1.2CP3CPCheckpoint 4 Worked-out solution available at LarsonAppliedCalculus.com Find the rate of change of the revenue with respect to time for the company in Example 4 when the weekly demand function is p=1500.002x.1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWUIn Exercises 7-10, find dy/dx by implicit differentiation. x34y+2xy=12xIn Exercises 7-10, find dy/dx by implicit differentiation. x+xy2y2=xyUsing Related Rates In Exercises 14, assume that x and y are both differentiable functions of t. Use the given values to find (a) dy/dt and (b) dx/dt. See Example 1. EquationFindGiveny=x(a)dydtwhenx=4,dxdt=3(b)dxdtwhenx=25,dydt=22EUsing Related Rates In Exercises 14, assume that x and y are both differentiable functions of t. Use the given values to find (a) dy/dt and (b) dx/dt. See Example 1. EquationFindGivenxy=4(a)dydtwhenx=8,dxdt=10(b)dxdtwhenx=1,dydt=6Using Related Rates In Exercises 14, assume that x and y are both differentiable functions of t. Use the given values to find (a) dy/dt and (b) dx/dt. See Example 1. EquationFindGivenx2+y2=25(a)dydtwhenx=3,y=4,dxdt=8(b)dxdtwhenx=4,y=3,dydt=2Area The radius r of a circle is increasing at a rate of 3 inches per minute. Find the rate of change of the area when (a) r = 6 inches and (b) r = 24 inches.6E7EVolume Let V be the volume of a sphere of radius r that is changing with respect to time. If dr/dt is constant, is dV/dt constant? Explain your reasoning.9E10ECost, Revenue, and Profit A company that manufactures sport supplements calculates that its cost C and revenue R can be modeled by the equations C=125,000+0.75xandR=250x110x2 where x is the number of units of sport supplements produced in 1 week. Production during one particular week is 1000 units and is increasing at a rate of 150 units per week. Find the rate of change of the (a) cost, (b) revenue, and (c) profit.Cost, Revenue, and Profit A company that manufactures pet toys calculates that its cost C and revenue R can be modeled by the equations C=75,000+1.05xandR=500xx225 where x is the number of toys produced in 1 week. Production during one particular week is 5000 toys and is increasing at a rate of 250 toys per week. Find the rate of change of the (a) cost, (b) revenue, and (c) profit.Revenue The revenue R from selling x units of a product is given by R=1200xx2. The sales are increasing at a rate of 23 units per day. Find the rate of change of the revenue when (a) x = 300 units and (b) x = 450 units.Profit The profit P from selling x units of a product is given by P = 510x 0.3x2. The sales are increasing at a rate of 9 units per day. Find the rate of change of the profit when (a) x = 400 units and (b) x = 600 units.15E16EBoating A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat (see figure). The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 13 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock?Shadow 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?Air traffic Control An airplane flying at an altitude of 6 miles passes directly over a radar antenna (see figure). When the airplane is 10 miles away (s = 10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the airplane?Baseball A (square) baseball diamond has sides that are 90 feet long (see figure). A player running from second base to third base at a speed of 30 feet per second is 26 feet from third base. At what rate is the players distance from home plate changing?Air Traffic Control An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 450 miles per hour. The other is 200 miles from the point and has a speed of 600 miles per hour. (a) At what rate is the distance between the planes changing? (b) How much time does the controller have to get one of the airplanes on a different flight path?Advertising Costs A retail sporting goods store estimates that weekly sales S and weekly advertising costs x are related by the equation S = 2250 + 50x + 0.35x2. The current weekly advertising costs are $1500, and these costs are increasing at a rate of $125 per week. Find the current rate of change of the weekly sales with respect to time.Environment An accident at an oil drilling platform is causing a circular oil slick. The slick is 0.08 foot thick, and when the radius of the slick is 150 feet, the radius is increasing at the rate of 0.5 foot per minute. At what rate (in cubic feet per minute) is oil flowing from the site of the accident?24ESales The profit for a product is increasing at a rate of $5600 per week. The demand and cost functions for the product are given by p = 6000 25x and C = 2400x + 5200, where x is the number of units produced per week. Find the rate of change of the sales with respect to time when the weekly sales are x = 44 units.26EFinding Critical numbers In Exercises 16, find the critical numbers of the function. f(x)=x25x+9Finding Critical numbers In Exercises 16, find the critical numbers of the function. y=3x2+18x3RE4RE5RE