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All Textbook Solutions for Calculus (MindTap Course List)
If f and g are the functions whose graphs are shown, let u(x)=f(g(x)),v(x)=g(f(x)),, and w(x)=g(g(x)). Find each derivative, if it exists. If it does not exist, explain why. a u1 b v1 c w1If f is the function whose graph is shown, let h(x)=f(f(x)) and g(x)=f(x2). Use the graph of f to estimate the value of each derivative. a h2 b g2If g(x)=f(x), where the graph of f is shown, evaluate g3.68E69E70E71E72EFind the given derivative by finding the first few derivatives and observing the pattern that occurs. D103cos2x74EThe displacement of a particle on a vibrating string is given by the equation s(t)=10+14sin(10t) where .s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.76E77E78E79EAir is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is Vt and its radius is rt. a What do the derivatives dV/dr and dV/dt represent? b Express dV/dt in terms of dr/dt.81E82EUse the Chain Rule to prove the following. a The derivative of an even function is an odd function. b The derivative of an odd function is an even function.Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule. Hint: Write f(x)/g(x)=f(x)[g(x)1.].85E86E87Ea Write |x|=x2 and use the Chain Rule to show that b If f(x)=|sinx|, find fx and sketch the graphs of f and f. Where is f not differentiable? c If g(x)=sin|x|, find g'x and sketch the graphs of g and g'. Where is g not differentiable?If y=f(u) and u=g(x), where f and g are twice differentiable functions, show that d2dx2=d2du2(dudx)2+dydud2udx290Ea Find y by implicit differentiation. b Solve the equation explicitly for y and differentiate to get y in terms of x. c Check that your solutions to parts a and b are consistent by substituting the expression for y into your solution for part a. 9x2y2=1a Find y by implicit differentiation. b Solve the equation explicitly for y and differentiate to get y in terms of x. c Check that your solutions to parts a and b are consistent by substituting the expression for y into your solution for part a. 2x2+x+xy=1a Find y by implicit differentiation. b Solve the equation explicitly for y and differentiate to get y in terms of x. c Check that your solutions to parts a and b are consistent by substituting the expression for y into your solution for part a. x+y=1a Find y by implicit differentiation. b Solve the equation explicitly for y and differentiate to get y in terms of x. c Check that your solutions to parts a and b are consistent by substituting the expression for y into your solution for part a. 2x1y=4Find dy/dx by implicit differentiation. x24xy+y2=4Find dy/dx by implicit differentiation. 2x2+xyy2=27EFind dy/dx by implicit differentiation. x3xy2+y3=1Find dy/dx by implicit differentiation. x2x+y=y2+1Find dy/dx by implicit differentiation. y5+x2y3=1+x4yFind dy/dx by implicit differentiation. ycosx=x2+y212EFind dy/dx by implicit differentiation. x+y=x4+y414EFind dy/dx by implicit differentiation. tan(x/y)=x+yFind dy/dx by implicit differentiation. xy=x2+y2Find dy/dx by implicit differentiation. xy=1+x2yFind dy/dx by implicit differentiation. xsiny+ysinx=1Find dy/dx by implicit differentiation. sin(xy)=cos(x+y)20EIf f(x)+x2[f(x)]3=10 and f(1)=2, find f(1).If g(x)+xsing(x)=x2, find g(0).Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. x4y2x3y+2xy3=0Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. ysecx=xtanyUse implicit differentiation to find an equation of the tangent line to the curve at the given point. ysin2x=xcos2y, (/2,/4)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. sin(x+y)=2x2y, (,)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2xyy2=1, 2, 1 hyperbolaUse implicit differentiation to find an equation of the tangent line to the curve at the given point. x2+2xy+4y2=12, 2, 1 ellipseUse implicit differentiation to find an equation of the tangent line to the curve at the given point. x2+y2=(2x2+2y2x)2, (0,12) cardioid30EUse implicit differentiation to find an equation of the tangent line to the curve at the given point. 2(x2+y2)2=25(x2y2), 3, 1 lemniscusesUse implicit differentiation to find an equation of the tangent line to the curve at the given point. y2(y24)=x2(x25), (0,2) devils curvea The curve with equation y2=5x4x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point 1, 2. b Illustrate part a by graphing the curve and the tangent line on a common screen. If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.a The curve with equation y2=x2+3x2 is called the Tschimhausen cubic. Find an equation of the tangent line to this curve at the point (1,2). b At what points does this curve have horizontal tangents? c Illustrate parts a and b by graphing the curve and the tangent lines on a common screen.Find y by implicit differentiation. x2+4y2=4Find y by implicit differentiation. x2+xy+y2=3Find y by implicit differentiation. siny+cosx=1Find y by implicit differentiation. x3y3=7If xy+y3=1, find the value of y at the point where x=0.40EFanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. a Graph the curve with equation y(y21)(y2)=x(x1)(x2) At how many points does this curve have horizontal tangents? Estimate the x-coordinates of these points. b Find equations of the tangent lines at the points 0, 1 and 0, 2. c Find the exact x-coordinates of the points in part a. d Create even more fanciful curves by modifying the equation in part a.a The curve with equation 2y3+y2y5=x42x3+x2 has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why. b At how many points does this curve have horizontal tangent lines? Find the x-coordinates of these points.43EShow by implicit differentiation that the tangent to the ellipse x2a2+y2b2=1 at the point (x0.,y0) is x0xa2+y0yb2=1Find an equation of the tangent line to the hyperbola x2a2y2b2=1 at the point (x0,y0).46EShow, using implicit differentiation, that any tangent fine at a point P to a circle with center O is perpendicular to the radius OP.48ETwo curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. x2+y2=r2, ax+by=0Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. x2+y2=ax, x2+y2=byTwo curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. y=cx2, x2+2y2=kTwo curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. y=ax2, x2+3y2=bShow that the ellipse x2/a2+y2/b2=1 and the hyperbola x2/A2y2/B2=1 are orthogonal trajectories if A2a2 and a2b2=A2+B2 so the ellipse and hyperbola have the same foci.54Ea The van der Waals equation for n moles of a gas is (P+n2aV2)(Vnb)=nRT where P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas. If T remains constant, use implicit differentiation to find dV/dP. b Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V =10 L and a pressure of P = 2.5 atm. Use a = 3.592 L2-atm/mole2 and b = 0.04267 L/mole.56EThe equation x2xy+y2=3 represents a rotated ellipse, that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.58EFind all points on the curve x2y2+xy=2 where the slope of the tangent line is 1.Find equations of both the tangent lines to the ellipse x2+4y2=36 that pass through the point 12, 3.The Bessel function of order 0, y=J(x) satisfies the differential equation xy+y+xy=0 for all values of x and its value at 0 is J0 = 1. a Find J0. b Use implicit differentiation to find J0.The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2+4y25. If the point (5,0) is on the edge of the shadow, how far above the x-axis is the lamp located?A particle moves according to a law of motion s=f(t),t0, where t is measured in seconds and s in feet. a Find the velocity at time t. b What is the velocity after 1 second? c When is the particle at rest? d When is the particle moving in the positive direction? e Find the total distance traveled during the first 6 seconds. f Draw a diagram like Figure 2 to illustrate the motion of the particle. g Find the acceleration at time t and after 1 second. h Graph the position, velocity, and acceleration functions for 0t6. i When is the particle speeding up? When is it slowing down? f(t)=t39t2+24tA particle moves according to a law of motion s=f(t),t0, where t is measured in seconds and s in feet. a Find the velocity at time t. b What is the velocity after 1 second? c When is the particle at rest? d When is the particle moving in the positive direction? e Find the total distance traveled during the first 6 seconds. f Draw a diagram like Figure 2 to illustrate the motion of the particle. g Find the acceleration at time t and after 1 second. h Graph the position, velocity, and acceleration functions for 0t6. i When is the particle speeding up? When is it slowing down? f(t)=0.01t4+0.04t3A particle moves according to a law of motion s=f(t),t0, where t is measured in seconds and s in feet. a Find the velocity at time t. b What is the velocity after 1 second? c When is the particle at rest? d When is the particle moving in the positive direction? e Find the total distance traveled during the first 6 seconds. f Draw a diagram like Figure 2 to illustrate the motion of the particle. g Find the acceleration at time t and after 1 second. h Graph the position, velocity, and acceleration functions for 0t6. i When is the particle speeding up? When is it slowing down? f(t)=sin(t/2)A particle moves according to a law of motion s=f(t),t0, where t is measured in seconds and s in feet. a Find the velocity at time t. b What is the velocity after 1 second? c When is the particle at rest? d When is the particle moving in the positive direction? e Find the total distance traveled during the first 6 seconds. f Draw a diagram like Figure 2 to illustrate the motion of the particle. g Find the acceleration at time t and after 1 second. h Graph the position, velocity, and acceleration functions for 0t6. i When is the particle speeding up? When is it slowing down? f(t)=9tt2+9Graphs of the velocity functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.Graphs of the position functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.The height in meters of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h=2+24.5t4.9t2 after t seconds. a Find the velocity after 2 s and after 4 s. b When does the projectile reach its maximum height? c What is the maximum height? d When does it hit the ground? e With what velocity does it hit the ground?If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s=80t16t2. a What is the maximum height reached by the ball? b What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?If a rock is thrown vertically upward from the surface of Mars with velocity 15 m/s, its height after t seconds is h=15t1.86t2. a What is the velocity of the rock after 2 s? b What is the velocity of the rock when its height is 25 m on its way up? On its way down?10Ea A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area Ax of a wafer changes when the side length x changes. Find A15 and explain its meaning in this situation. b Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount x. How can you approximate the resulting change in area A if x is small?a Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning. b Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11b.a Find the average rate of change of the area of a circle with respect to its radius r as r changes from i 2 to 3 ii 2 to 2.5 iii 2 to 2.1 b Find the instantaneous rate of change when r = 2. c Show that the rate of change of the area of a circle with respect to its radius at any r is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount r. How can you approximate the resulting change in area A if r is small?A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after a 1 s, b 3 s, and c 5 s. What can you conclude?A spherical balloon is being inflated. Find the rate of increase of the surface area (S=4r2) with respect to the radius r when r is a 1 ft, b 2 ft, and c 3 ft. What conclusion can you make?16EThe mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x2 kg. Find the linear density see Example 2 when x is a 1 m, b 2 m, and c 3 m. Where is the density the highest? The lowest?If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricellis Law gives the volume V of water remaining in the tank after t minutes as V=5000(1+140t)20t40 Find the rate at which water is draining from the tank after a 5 min, b 10 min, c 20 min, and d 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.The quantity of charge Q in coulombs C that has passed through a point in a wire up to time t measured in seconds is given by Q(t)=t32t2+6t+2. Find the current when a t = 0.5 s and b t = 1 s. See Example 3. The unit of current is an ampere 1 A = 1 C/s. At what time is the current lowest?20E21ESome of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at 6:45 am. This helps explain the following model for the water depth D in meters as a function of the time t in hours after midnight on that day: D(t)=7+5cos[0.503(t6.75)] How fast was the tide rising or falling at the following times? a 3:00 am b 6:00 am c 9:00 am d NoonBoyles Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV=C a Find the rate of change of volume with respect to pressure. b A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. c Prove that the isothermal compressibility see Example 5 is given by =1/P.24EThe table gives the population of the world Pt, in millions, where t is measured in years and t = 0 corresponds to the year 1900. t Population millions 0 1650 10 1750 20 1860 30 2070 40 2300 50 2560 60 3040 70 3710 80 4450 90 5280 100 6080 110 6870 a Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant fines. b Use a graphing device to find a cubic function a third- degree polynomial that models the data. c Use your model in part b to find a model for the rate of population growth. d Use part c to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part a. e Estimate the rate of growth in 1985.26E27E28E29EThe cost function for a certain commodity is C(q)=84+0.16q0.0006q2+0.000003q3 a Find and interpret C 100. b Compare C100 with the cost of producing the 101st item.If px is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is A(x)=p(x)x a Find Ax. Why does the company want to hire more workers if ax0? b Show that ax 0if p'x is greater than the average productivity.32EThe gas law for an ideal gas at absolute temperature T in kelvins, pressure P in atmospheres, atm, and volume V in liters is PV=nRT, where n is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at a certain instant, P = 8.0 atm and is increasing at a rate of 0.10 atm/min and V = 10 F and is decreasing at a rate of 0.15 L/min. Find the rate of change of T with respect to time at that instant if n = 10 moles.Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f(r)=2Dr, where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate r and explain its meaning.In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by Wt, and caribou, given by Ct, in northern Canada. The interaction has been modeled by the equations dCdt=aCbCWdWdt=cW+dCW a What values of dC/dt and dW/dt correspond to stable populations? b How would the statement The caribou go extinct be represented mathematically? c Suppose that a = 0.05, b = 0.001, c = 0.05, and d = 0.0001. Find all population pairs C, W that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation dPdt=r0(1P(t)Pc)P(t)+P(t) where r0 is the birth rate of the fish, Pc, is the maximum population that the pond can sustain called the carrying capacity, and is the percentage of the population that is harvested. a What value of dP/dt corresponds to a stable population? b If the pond can sustain 10, 000 fish, the birth rate is 5, and the harvesting rate is 4, find the stable population level. c What happens if is raised to 5?1Ea If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt. b Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30 m?Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16cm2?4EA cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/min. How fast is the height of the water increasing?6EThe radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 8 cm?The area of a triangle with sides of lengths a and b and contained angle is A=12absin a If a = 2 cm, b = 3 cm, and 0 increases at a rate of 0.2 rad/ min, how fast is the area increasing when =3? b If a = 2 cm, b increases at a rate of 1.5 cm/min, and increases at a rate of 0.2 rad/ min, how fast is the area increasing when b = 3 cm and =3? c If a increases at a rate of 2.5 cm/min, b increases at a rate of 1.5 cm/min, and increases at a rate of 0.2 rad/min, how fast is the area increasing when a = 2 cm, b = 3 cm, and =3?Suppose y=2x+1, where x and y are functions of t. a If dx/dt= 3, find dy/dt when x=4. b If dy/dt= 5, find dx/dt when x=12.10E11EA particle is moving along a hyperbola xy=8. As it reaches the point 4, 2, the y-coordinate is decreasing at a rate of 3 cm/s. How fast is the x-coordinate of the point changing at that instant?13-16 a What quantities are given in the problem? b What is the unknown? c Draw a picture of the situation for any time t. d Write an equation that relates the quantities. e Finish solving the problem. A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.13-16 a What quantities are given in the problem? b What is the unknown? c Draw a picture of the situation for any time t. d Write an equation that relates the quantities. e Finish solving the problem. If a snowball melts so that its surface area decreases at a rate of 1 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm.13-16 a What quantities are given in the problem? b What is the unknown? c Draw a picture of the situation for any time t. d Write an equation that relates the quantities. e Finish solving the problem. A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?16ETwo cars start moving from the same point. One travels south at 60 mi /h and the other travels west at 25 mi / h. At what rate is the distance between the cars increasing two hours later?18EA man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking?A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. a At what rate is his distance from second base decreasing when he is halfway to first base? b At what rate is his distance from third base increasing at the same moment?The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100cm2?A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 pm?24EWater is leaking out of an inverted conical tank at a rate of 10,000cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12ft3/min, how fast is the water level rising when the water is 6 inches deep?A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2m3/min, how fast is the water level rising when the water is 30 cm deep?A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross- section is shown in the figure. If the pool is being filled at a rate of 0.8ft3/min, how fast is the water level rising when the depth at the deepest point is 5 ft?Gravel is being dumped from a conveyor belt at a rate of 30ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?31E32EThe top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?If the minute hand of a clock has length r in centimeters, find the rate at which it sweeps out area as a function of r.36E37EWhen air expands adiabatically without gaining or losing heat, its pressure P and volume V are related by the equation PV1.4=C, where C is a constant. Suppose that at a certain instant the volume is 400cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at this instant?39EBrain weight B as a function of body weight W in fish has been modeled by the power function B=0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L measured in centimeters is W=0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species brain growing when the average length was 18 cm?41ETwo carts, A and B, are connected by a rope 39 ft long that passes over a pulley P see the figure. The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?43E44EA plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is /3, this angle is decreasing at a rate of /6 radians per minute. How fast is the plane traveling at that time?A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance from the plane to the radar station increasing a minute later?Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runners friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one oclock?14 Find the linearization L(x) of the function at a. f(x)=x3x2+3,a=-22E14 Find the linearization L(x) of the function at a. f(x)=x,a=44E5EFind the linear approximation of the function g(x)=1+xata=0 at a=0 and use it to approximate the numbers 0.953 and 1.13. Illustrate by graphing g and the tangent line.7E8E7-10 Verify the given linear approximation at a=0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 1/(1+2x)418x10E11-14 Find the differential dy of each function. a y=(x23)2 b y=1t411-14 Find the differential dy of each function. a y=1+2u1+3u b y=2sin211-14 Find the differential dy of each function. a y=tant b y=1v21+v214E15-18 a Find the differential dy and b evaluate dy for the given values of x and dx.y=tanx,x=/4,dx=0.116E17E18E19-22 Compute y and dy for the given values of x and dx=x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. y=x24x,x=3,x=0.520E19-22 Compute y and dy for the given values of x and dx=x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. y=x2,x=3,x=0.822E23-28 Use a linear approximation or differentials to estimate the given number. (1.999)424E25E23-28 Use a linear approximation or differentials to estimate the given number. 100.527E28E29-30 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. sec0.08130EThe edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing a the volume of the cube and b the surface area of the cube.The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. a Use differentials to estimate the maximum error in the calculated area of the disk. b What is the relative error? What is the percentage error?The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. a Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? b Use differentials to estimate the maximum error in the calculated volume. What is the relative error?Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.a Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness r. b What is the error involved in using the formula from part a?One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30, with a possible error of 1. a Use differentials to estimate the error in computing the length of the hypotenuse. b What is the percentage error?37EWhen blood flows along a blood vessel, the flux F the volume of blood per unit time that flows past a given point is proportional to the fourth power of the radius R of the blood vessel: F=kR4 This is known as Poiseuilles Law; we will show why it is true in Section 8.4. A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5 increase in the radius affect the flow of blood?Establish the following rules for working with differentials where c denotes a constant and u and v are functions of x. a dc=0 b d(cu)=cdu c d(u+v)=du+dv d d(uv)=udv+vdu e (uv)=vduudvv2 f d(xn)=nxn1dx40ESuppose that the only information we have about a function f is that f(1)=5 and the graph of its derivative is as shown. a Use a linear approximation to estimate f(0.9) and f(1,1) b Are your estimates in part a too large or too small? Explain.Suppose that we dont have a formula for g(x) but we know that g(2)=4andg'(x)=x2+5 for all x. a Use a linear approximation to estimate g(1.95) and g(2.05). b Are your estimates in part a too large or too small? Explain.Write an expression for the slope of the tangent line to the curve y=f(x) at the point (a,f(a)).2CC3CCDefine the derivative fa. Discuss two ways of interpreting this number.5CC6CC7CC8CC9CC10CCGive several examples of how the derivative can be interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences.12CCDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous at a then f is differentiable at a.2TFQ3TFQ4TFQ5TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is differentiable, then ddxf(x)=f(x)2x.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. ddx|x2+x|=|2x+1|8TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If g(x)=x5, then limx2g(x)g(2)x2=8010TFQ11TFQ12TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a polynomial is a polynomial.14TFQ15TFQThe displacement in meters of an object moving in a straight line is given by s=1+2t+14t2, where t is measured in seconds. a Find the average velocity over each time period. i [1,3] ii [1,2] iii [1,1.5] iv [1,1.1] b Find the instantaneous velocity when t=1.The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.3E4EThe figure shows the graph of f,f, and f".Identify each curve, and explain your choice.6EThe total cost of repaying a student loan at an interest rate of r per year is C=f(r). a What is the meaning of the derivative f(r)? What are its units? b What does the statement f(10)=1200 mean? c Is f(r) always positive or does it change sign?8E9E10E11E12E13E14E1340 Calculate y. y=x2x+2x16E17E18E19E20E21E22E23E24E25E26E27E28E1340 Calculate y. sin(xy)=x2y30E31E32E33E34E35E36E37E38E39E40EIf f(t)=4t+1, find f"(2).42EFind y" if x6+y6=1.44E45E46E47E48E49E50E51E52E53E54E55EHow many tangent fines to the curve y=x/(x+1) pass through the point 1, 2? At which points do these tangent fines touch the curve?If f(x)=(xa)(xb)(xc), show that f(x)f(x)=1xa+1xb+1xc