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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 76.f(x)=x2/3,forx077E78ETowing a boat A boat is towed toward a dock by a cable attached to a winch that stands 10 feet above the water level (see figure). Let be the angle of elevation of the winch and let be the length of the cable as the boat is towed toward the dock. a. Show that the rate of change of with respect to is dd=102100. b. Compute ddwhen = 50, 20 and 11 ft. c. Find lim10+ddand explain what happens as the last foot of cable is reeled in (note that the boat is at the dock when = 10). d. It is evident from the figure that increases as the boat is towed to the dock. Why, then, is d/dnegative?Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45 angle from the horizontal (see figure). a. Express the angle of elevation from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80 h.) b. What is the rate of change of with respect to the birds height when it is 60 feet above the ground?Angle to a particle, part I A particle travels clockwise on a circular path of diameter D, monitored by a sensor on the circle at point P; the other endpoint of the diameter on which the sensor lies is Q (see figure). Let be the angle between the diameter PQ and the line from the sensor to the particle. Let c be the length of the chord from the particles position to Q. a. Calculate d/dc. b. Evaluate ddc|c=0.Angle to a particle (part 2) The figure in Exercise 81 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers. Angle to a particle (part 1) A particle travels clockwise on a circular path of diameter D, monitored by a sensor on the circle at point P; the other endpoint of the diameter on which the sensor lies is Q (see figure). Let be the angle between the diameter PQ and the line from the sensor to the particle. Let c be the length of the chord from the particles position to Q. a. Calculate ddc. b. Evaluate ddc|c=0Derivative of the inverse sine Find the derivative of the inverse sine function using Theorem 3.21.Derivative of the inverse cosine Find the derivative of the inverse cosine function in the following two ways. a. Using Theorem 3.21 b. Using the Identity sin1+cos1x=/285E86EIdentity proofs Prove the following identities and give the values of x for which they are true. 75. cos(sin1x)=1x2Identity proofs Prove the following identities and give the values of x for which they are true. 76. cos(2sin1x)=12x2Identity proofs Prove the following identities and give the values of x for which they are true. 77. tan(2tan1x)=2x1x290EIn Example 1, what is the rate of change of the area when the radius is 200 m? 200 m? Example 1 Spreading Oil An oil rig springs a leak in calm seas, and the oil spreads in a circular patch around the rig. If the radius or the oil patch increases at a rate of 30m/hr, how fast is the area of the patch increasing when the patch has a radius of 100 meters (Figure 3.77)?Assuming the same pane speeds as In Example 2, how fast is the distance between the planes changing if x = 60 mi and y = 75 mi? Example 2 Converging Airplanes Two small planes approach an airport, one flying due west a: 120 mi/hr and the other flying due north at 150 mi/hr. Assuming they fly at the same constant elevation, how fast is the distance between the planes changing when the westbound plane is 180 miles from the airport and the northbound plane is 225 miles from the airport?In Example 3, what is the rate of change of the height when h = 2 in? Example 3 Morning Coffee Coffee is draining out of a conical filter at a rate of 2.25 in3/min. If the core is 5 in tall and has a radius of 2 in, how fast is the coffee level dropping when the coffee is 3 in deep?In Example 4, notice that as the balloon rises (as increases), the rate of change of the angle of elevation decreases to zero. When does the maximum value of (t) occur, and what is it? Example 4 Observing a Launch An observer stands 200 meters from the launch site of a hot-air balloon at an elevation equal to the elevation of the launch site. The balloon rises vertically at a constant rate of 4 m/s. How fast is the angle of elevation of the balloon increasing 30 seconds after the launch? (The angle of elevation is the angle between the ground and the observers line of sight to the balloon.)Give an example in which one dimension of a geometric figure changes and produces a corresponding change in the area or volume of the figure.Charles law states that for a fixed mass of gas under constant pressure, the volume V and temperature T of the gas (in kelvins) satisfy the equal on V = kT, where k is constant. Find an equation relating dV/dt to dT/dt.If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?The temperature F in degrees Fahrenheit is related to the temperature C in degrees Celsius by the equation F=95C+32. a. Find an equation relating dF/dt to dC/dt. b. How fast s the temperature in an oven changing in degrees Fahrenheit per minute if it is rising at 10 Celsius per min?A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water. a. If t is elapsed time, h is the height of the water, and V is the volume of the water, find equations relating V to h and dV/dt to dh/dt. b. At what rate is the volume of the water increasing if the water level is rising at 14ft/min? c. At what rate is the water level rising if the pool is filed at a rate of 10 ft3/min?At all times, the length of a rectangle is twice the width w of the rectangle as the area of the rectangle changes with respect to time t. a. Find an equation relating A to w. b. Find an equation relating dA/dt to dW/dt.The volume V of a sphere of radius r changes over time t. a. Find an equation relating dV/dt to dr/dt. b. At what rate is the volume changing if the radius increases at 2 in/min when the radius is 4 inches? c. At what rate is the radius changing if the volume increases at 10 in3/min when the radius is 5 inches?At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dX/dt.Assume x, y, and z are functions of t with z=x+y3. Find dz/dt=1, dy/dt=5, and y = 2.10EExpanding square The sides of a square increase in length at a rate of 2 m/s. a. At what rate is the area of the square changing when the sides are 10 m long? b. At what rate is the area of the square changing when the sides are 20 m long?Shrinking square The sides of a square decrease in length at a rate of 1 m/s. a. At what rate is the area of the square changing when the sides are 5 m long? b. At what rate are the lengths of the diagonals of the square changing?Expanding isosceles triangle The legs of an isosceles right triangle increase in length at a rate of 2 m/s. a. At what rate is the area of the triangle changing when the legs are 2 m long? b. At what rate is the area of the triangle changing when the hypotenuse is 1 m long? c. At what rate is the length of the hypotenuse changing?Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s. a. At what rate is the area of the triangle changing when the legs are 5 m long? b. At what rate are the lengths of the legs of the triangle changing? c. At what rate is the area of the triangle changing when the area is 4 m2?Expanding circle The area of a circle increases at a rate of 1 cm2/s. a. How fast is the radius changing when the radius is 2 cm? b. How fast is the radius changing when the circumference is 2 cm?16EShrinking circle A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?18EBalloons A spherical balloon is inflated and its volume increases at a rate of 15 in3/min. What is the rate of change of its radius when the radius is 10 in?Expanding rectangle A rectangle initially has dimensions 2 cm by 4 cm. All sides begin increasing in length at a rate of 1 cm/s. At what rate is the area of the rectangle increasing after 20 s?Melting snowball A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area 4r2.)Divergent paths Two beats leave a pert at the same time; one travels west at 20 mi/hr and the other travels south at 15 mi/hr. a. After 30 minutes, how far is each boat from port? b. At what rate is the distance between the boats changing 30 minutes after they leave the port?Time-lagged flights An airliner passes over an airport at noon traveling 500 mi/hr due west. At l:00 P.M., another airliner passes over the same airport at the same elevation traveling due north at 550 mi/hr. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 2:30 P.M.?Flying a kite Once Kates kite reaches a height of 50 ft (above her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kates hands at the moment that she has released 120 ft of string?Rope on a boat A rope passing through a capstan on a dock is attached to a boat offshore. The rope is pulled in at a constant rate of 3 ft/s, and the capstan is 5 ft vertically above the water. How fast is the boat traveling when it is 10 ft from the dock?Bug on a parabola A bug is moving along the right side of the parabola y=x2 at a rate such that its distance from the origin is increasing at 1 cm/min. a. At what rate is the x-coordinate of the bug increasing at the point (2, 4)? b. Use the equation y = x2 to find an equation relating dydt to dxdt c. At what rate is the y-coordinate of the bug increasing at the point (2, 4)?Another balloon story A hot-air balloon is 150 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 40 mi/hr (58.67 ft/s). If the balloon rises vertically at a rate of 10 ft/s, what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?Baseball runners Runners stand at first and second base in a baseball game. At the moment a ball is hit, the runner at first base runs to second base at 18 ft/s; simultaneously, the runner on second runs to third base at 20 ft/s. How fast is the distance between the runners changing 1 second after the ball is hit (see figure)? (Hint: The distance between consecutive bases is 90 ft and the bases lie at the comers of a square.)Another fishing story An angler hooks a trout and reels in his line at 4 in/s. Assume the tip of the fishing rod is 12 ft above the water and directly above the angler, and the fish is pulled horizontally directly toward the angler (see figure). Find the horizontal speed of the fish when it is 20 ft from the angler.Parabolic motion An arrow is shot into the air and moves along the parabolic path y = x(50 x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (i) x = 10 and (ii) x = 40?Draining a water heater A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft3/min) if the water level is dropping at 6 in/min?Drinking a soda At what rate is soda being sucked out of a cylindrical glass that is 6 in tall and has a radius of 2 in? The depth of the soda decreases at a constant rate of 0.25 in/s.Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber?Filling two pools Two cylindrical swimming pools are being filled simultaneously at the same rate (in m3/min; see figure). The smaller pool has a radius of 5 m, and the water level rises at a rate of 0.5 m/min. The larger pool has a radius of 8 m. How fast is the water level rising in the larger pool?Growing sandpile Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three times its height. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 12 cm high. At what rate is the sand leaving the bin at that instant?Draining a tank An inverted conical water tank with a height of 12 ft and a radius of 6 ft is drained through a hole in the vertex at a rate of 2 ft3/s (see figure). What is the rate of change of the water depth when the water depth is 3 ft? (Hint: Use similar triangles.)37ETwo tanks A conical tank with an upper radius of 4 m and a height of 5 m drains into a cylindrical tank with a radius of 4 m and a height of 5 m (see figure). If the water level in the conical tank drops at a rate of 0.5 m/min, at what rate does the water level in the cylindrical tank rise when the water level in the conical tank is 3 m? 1 m?Filling a hemispherical tank A hemispherical tank with a radius of 10 m is filled from an inflow pipe at a rate of 3 m3/min (see figure). How fast is the water level rising when the water level is 5 m from the bottom of the tank? (Hint: The volume of a cap of thickness h sliced from a sphere of radius r is h2(3r h)/3.)Surface area of hemispherical tank Per the situation described in Exercise 39, what is the rate of change of the area of the exposed surface of the water when the water is 5 m deep? Filling a hemispherical tank A hemispherical tank with a radius of 10 m is filed from an inflow pipe at a rate of 3 m3/min (see figure). How fast is the water level rising when the water level is 5 m from the bottom of the tank? (Hint: The volume of a cap of thickness h sliced from a sphere of radius r is h2(3rh)/3.)Ladder against the wall A 13-foot ladder is leaning against a vertical wall (see figure) when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.5 ft/s. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 5 ft from the wall?Ladder against the wall again A 12-foot ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant that the vertical speed of the top of the ladder equals the horizontal speed of the foot of the Ladder?Moving shadow A 5-foot-tall woman walks at 8 ft/s toward a streetlight that is 20 ft above the ground. What is the rate of change of the length of her shadow when she is 15 ft from the streetlight? At what rate is the tip of her shadow moving?Another moving shadow A landscape light at ground level lights up the side of a tall building that is 15 feet from the light. A 6-ft-tall man starts walking (on flat terrain) from the light directly toward the building. How fast is he walking when he is 9 feet from the light if his shadow on the building is shrinking at 2 ft/s at that instant?Watching an elevator An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observers line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground? When the elevator is 40 m above the ground?Observing a launch An observer stands 300 ft from the launch site of a hot-air balloon. The balloon is launched vertically and maintains a constant upward velocity of 20 ft/s. What is the rate of change of the angle of elevation of the balloon when it is 400 ft from the ground? The angle of elevation is the angle between the observers line of sight to the balloon and the ground.Viewing angle The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?Altitude of a jet A jet ascends at a 10 angle from the horizontal with an airspeed of 550 mi/hr (its speed along its line of flight is 550 mi/hr). How fast is the altitude of the jet increasing? If the sun is directly overhead, how fast is the shadow of the jet moving on the ground?Rate of dive of a submarine A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20 below the horizontal. How fast is the submarines altitude decreasing?A lighthouse problem A lighthouse stands 500 m off a straight shore and the focused beam of its light revolves four times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 200 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse.Filming a race A camera is set up at the starting line of a drag race 50 ft from a dragster at the starting line (camera 1 in the figure). Two seconds after the start of the race, the dragster has traveled 100 ft and the camera is turning at 0.75 rad/s while filming the dragster. a. What is the speed of the dragster at this point? b. A second camera (camera 2 in the figure) filming the dragster is located on the starting line 100 ft away from the dragster at the start of the race. How fast is this camera turning 2 seconds after the start of the race?Fishing reel An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the real (and the fishing line on it) is 2 inches. a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled In. Find an equation for L as a function of R. b. How fast is she reeling in her fishing line?Wind energy The kinetic energy E (in joules) of a mass in motion satisfies the equation E=12mv2, where mass m is measured in kg and velocity v is measured in m/s. a. Power P is defined to be dE/dt, the rate of change in energy with respect to time. Power is measured in units of watts (W), where 1 W = 1 joule/s. If the velocity v is constant, use implicit differentiation to find an equation for power P in terms of the derivative dm/dt. b. Wind turbines use kinetic energy in the wind to create electrical power. In this case, the derivative dm/dt is ca led the mass flow rate and it satisfies the equation dmdt=Av, where is the density of the air in kg/m3, A is the sweep area in m2 of the wind turbine (see figure), and v is the velocity of the wind in m/s. Show that P=12Av3. c. Suppose a blade on a small wind turbine has a length of 3 m. Find the available power P if the wind is blowing at 10 m/s. (Hint: Use =1.23kg/m3 for the density of air. The density of air varies, but this is a reasonable average value.) d. Wind turbines convert only a small percentage of the available wind power into electricity. Assume the wind turbine described in this exercise converts only 25% of the available wind power into electricity. How much electrical power is produced?Fishing reel An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the reel (and the fishing line on it) is 2 inches. a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled in. Find an equation for L as a function of R. b. How fast is she reeling in her fishing line?Clock hands The hands of the clock in the tower of the Houses of Parliament in London are approximately 3 m and 2.5 m in length. How fast is the distance between the tips of the hands changing at 9:00? (Hint: Use the Law of Cosines.)Divergent paths Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest at 15 mi/hr. At what rate is the distance between them changing 30 min after they leave the port?Filling a pool A swimming pool is 50 m long and 20 m wide. Its depth decreases linearly along the length from 3 m to 1 m (see figure). It was initially empty and is being filled at a rate of 1 m3/min. How fast is the water level rising 50 min after the filling begins? How long will it take to fill the pool?
Disappearing triangle An equilateral triangle initially has sides of length 20 ft when each vertex moves toward the midpoint of the opposite side at a rate of 1.5 ft/min. Assuming the triangle remains equilateral, what is the rate of change of the area of the triangle at the instant the triangle disappears?Oblique tracking A port and a radar station are 2 mi apart on a straight shore running east and west (see figure). A ship leaves the port at noon traveling northeast at a rate of 15 mi/hr. If the ship maintains its speed and course, what is the rate of change of the tracking angle between the shore and the line between the radar station and the ship at 12:30 P.M.? (Hint: Use the Law of Sines.)Oblique tracking A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.)61EWatching a Ferris wheel An observer stands 20 m from the bottom of a 10-m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of rad/min, and the observers line of sight with a specific seat on the wheel makes an angle with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of ? Assume the observers eyes are level with the bottom of the wheel.Draining a trough A trough in the shape of a half cylinder has length 5 m and radius 1 m. The trough is full of water when a valve is opened, and water flows out of the bottom of the trough at a rate of 1.5 m3/hr (see figure). (Hint: The area of a sector of a circle of a radius r subtended by an angle is r2 /2.) a. How fast is the water level changing when the water level is 0.5 m from the bottom of the trough? b. What is the rate of change of the surface area of the water when the water is 0.5 m deep?Searchlightwide beam A revolving searchlight, which is 100 m from the nearest point on a straight highway, casts a horizontal beam along a highway (see figure). The beam leaves the spotlight at an angle of /16 rad and revolves at a rate of /6 rad/s. Let w be the width of the beam as it sweeps along the highway and be the angle that the center of the beam makes with the perpendicular to the highway. What is the rate of change of w when = /3? Neglect the height of the searchlight.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function f(x) = |2x + 1| is continuous for all x; therefore, it is differentiable for all x. b. If ddx(f(x))=ddx(g(x)) then f = g. c. For any function f, ddxf(x)=f(x). d. The value of f(a) fails to exist only if the curve y = f(x) has a vertical tangent line at x = a. e. An object can have negative acceleration and increasing speed.Evaluate the derivative of each of the following functions using a limit definition of the derivation. Check your work by evaluating the derivatives using the derivatives rules given in this chapter. 2.f(x)=x2+2x+9Evaluate the derivative of each of the following functions using a limit definition of the derivation. Check your work by evaluating the derivatives using the derivatives rules given in this chapter. 3.g(x)1x2+5Evaluate the derivative of each of the following functions using a limit definition of the derivation. Check your work by evaluating the derivatives using the derivatives rules given in this chapter. 4.h(t)=3t+5Use differentiation to verify each equation. 5.ddx(tan3x3tanx+3x)=3tan4x.Use differentiation to verify each equation. 6.ddx(x1x2)=1(1x2)3/2.Use differentiation to verify each equation. 7.ddx(x4ln(x4+1))=4x71+x4.Use differentiation to verify each equation. 8.ddx(ln(1x1+x))=1x(x1).Evaluating derivatives Evaluate and simplify the following derivatives. 15. ddx(23x3+x2+7x+1)Evaluate and simplify y'. 10.y=4x4lnxx4Evaluate and simplify y'. 11.y=2xEvaluate and simplify y'. 12.y=2x2Evaluate and simplify y'. 13.y=e2Evaluate and simplify y'. 14.y=(2x3)x3/2Evaluate and simplify y'. 15.y=(1+x4)3/2Evaluating derivatives Evaluate and simplify the following derivatives. 16. ddx(2xx22x+2)Evaluating derivatives Evaluate and simplify the following derivatives. 17. ddt(5t2sint)Evaluating derivatives Evaluate and simplify the following derivatives. 18. ddx(5x+sin3x+sinx3)Evaluate and simplify y'. 19.y=ex(x2+2x+2)Evaluate and simplify y'. 20.y=lnxlnx+a, where a is constantEvaluate and simplify y'. 21.y=sec2wsec2w+1Evaluate and simplify y'. 22.y=(sinxcosx+1)1/3Evaluate and simplify y'. 23.y=ln|sec3x|Evaluate and simplify y'. 24.y=ln|csc7x+cot7x|Evaluate and simplify y'. 25.y=(5t2+10)100Evaluate and simplify y'. 26.y=esinx+2x+1Evaluate and simplify y'. 27.y=ln(sinx3)Evaluate and simplify y'. 28.y=etanx(tanx1)Evaluate and simplify y'. 29.y=tan1t21Evaluate and simplify y'. 30.y=xx+1Evaluating derivatives Evaluate and simplify the following derivatives. 19. dd(4tan(2+3+2))Evaluating derivatives Evaluate and simplify the following derivatives. 20. ddx(csc53x)Evaluate and simplify y'. 33.y=lnww5Evaluate and simplify y'. 34.y=seas, where a is a constantEvaluating derivatives Evaluate and simplify the following derivatives. 21. ddu(4u2+u8u+1)Evaluating derivatives Evaluate and simplify the following derivatives. 22. ddt(3t213t2+1)3Evaluating derivatives Evaluate and simplify the following derivatives. 23. dd(tan(sin))Evaluate and simplify y'. 38.y=(vv+1)4/3Evaluate and simplify y'. 39.y=sincos2x+1Evaluate and simplify y'. 40.y=esin(cosx)Evaluate and simplify y'. 41.y=lnet+1Evaluating derivatives Evaluate and simplify the following derivatives. 26. ddx(xe10x)Evaluate and simplify y'. 43.y=x2+2xtan1(cotx)Evaluate and simplify y'. 44.y=1x4+x2sin1x2Evaluating derivatives Evaluate and simplify the following derivatives. 27. ddx(xln2x)Evaluate and simplify y'. 46.y=e6xsinxEvaluating derivatives Evaluate and simplify the following derivatives. 29. ddx(2x2x)Evaluate and simplify y'. 48.y=10sinx+sin10xEvaluate and simplify y'. 49.y=(x2+1)lnxEvaluate and simplify y'. 50.y=xcos2xEvaluating derivatives Evaluate and simplify the following derivatives. 31. ddx(sin11x)Evaluating derivatives Evaluate and simplify the following derivatives. 30. ddx(log3(x+8))Evaluate and simplify y'. 53.y=6xcot13x+ln(9x2+1)Evaluate and simplify y'. 54.y=2x2cos1x+sin1xEvaluate and simplify y'. 55.x=cos(xy)Evaluate and simplify y'. 56.xy4+x4y=1Implicit differentiation Calculate y(x) for the following relations. 37. y=ey1+sinxImplicit differentiation Calculate y(x) for the following relations. 38. sinxcos(y1)=12Implicit differentiation Calculate y(x) for the following relations. 39. yx2+y2=15Evaluate and simplify y'. 60.y=(x2+1)3(x4+7)8(2x+1)7Evaluate and simplify y'. 61.y=(3x+5)10x2+5(x3+1)50Evaluating derivatives Evaluate and simplify the following derivatives. 34. f(1) when f(x) = tan1 (4x2)Evaluating derivatives Evaluate and simplify the following derivatives. 35. ddx(xsec1x)|x=23Evaluating derivatives Evaluate and simplify the following derivatives. 36. ddx(tan1ex)|x=0Evaluating derivatives Evaluate and simplify the following derivatives. 33. f(1) when f(x) = x1/xHigher-order derivatives Find and simplify y. 66.y=ex2+1Higher-order derivatives Find and simplify y. 67.y=2xxHigher-order derivatives Find and simplify y. 68.y=3x1x+1Higher-order derivatives Find and simplify y. 69.y=lnxx2Higher-order derivatives Find and simplify y. 70.x+siny=yHigher-order derivatives Find and simplify y. 71.xy+y2=1Tangent lines Find an equation of the line tangent to each of the following curves at the given point. 72.y=23x6;(2,1)Tangent lines Find an equation of the line tangent to each of the following curves at the given point. 73.y=3x3+sinx;(0,0)Tangent lines Find an equation of the line tangent to each of the following curves at the given point. 74.y=4xx2+3;(1,1)Tangent lines Find an equation of the line tangent to each of the following curves at the given point. 75.y+xy=6;(1,4)Tangent lines Find an equation of the line tangent to each of the following curves at the given point. 76.x2y+y3=5;(2,1)Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f, and g. 49. ddx(x2f(x))Derivative formulas Evaluate the fallowing derivatives. Express your answers in terms of f, g, f' and g'. 78.ddxf(x)g(x),f(x)g(x)0Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f, and g. 51. ddx(xf(x)g(x))Derivative formulas Evaluate the following derivatives. Express your answers in terms of f, g, f, and g. 52. ddxf(g(x)),g(x)0Matching functions and derivatives Match the functions in ad with the derivatives in AD.Sketching a derivative graph Sketch a graph of f for the function f shown in the figure.Sketching a derivative graph Sketch a graph of g for the function g shown in the figure.Use the given graphs of f and g to find each derivative. a. ddx(5f(x)+3g(x))|x=1 b. ddx(f(x)g(x))|x=1 c. ddx(f(x)g(x))|x=3 d. ddx(f(f(x)))|x=4 e. ddx(g(f(x)))|x=1Finding derivatives from a table Find the values of the following derivatives using the table. a. ddx(f(x)+2g(x))|x=3 b. ddx(f(x)g(x))|x=1 c. ddx(f(x)g(x))|x=3 d. ddx(f(x)3)|x=5 e. (g1)(7)Derivative of the inverse at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point. 56. f(x) = 1/(x + 1) at f(0)Derivative of the inverse at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point. 57. y=x3+x1aty=3Derivative of the inverse Find the derivative of the inverse of the following functions. Express the result with x as the independent variable. 58. f(x) = 12x 16Derivative of the inverse Find the derivative of the inverse of the following functions. Express the result with x as the independent variable. 59. f(x) = x1/3Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f. 62. a. ddx(xf(x))|x=2 b. ddx(f(x2))|x=1 c. ddx(f(f(x)))|x=1Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f. 63. a. (f1)(7) b. (f1)(3) c. (f1)(f(2))The line tangent to y=f(x) at x = 3 is y=4x10 and the line tangent to y=g(x) at x = 5 is y=6x27. 92.Compute f(3), f(3), g(5), and g(5).The line tangent to y=f(x) at x = 3 is y=4x10 and the line tangent to y=(g(x)) at x = 5 is y=6x27. 93.Find an equation of the line tangent to the graph of y=f(g(x)) at x = 5.Horizontal motion The position of an object moving horizontally after t seconds is given by the function s=27tt3, for t 0, where s is measured in feet, with s 0 corresponding to positions right of the origin. a. When is the object stationary, moving to the right, and moving to the left? b. Determine the velocity and acceleration of the object at t = 2. c. Determine the acceleration of the object when its velocity is zero. d. On what time intervals is the speed decreasing?Projectile on Mars Suppose an object is fired vertically upward from the ground on Mars with an initial velocity of 96 ft/s. The height s (in feet) of the object above the ground after t seconds is given by s=96t6t2. a. Determine the instantaneous velocity of the object at t = 1. b. When will the object have an instantaneous velocity of 12 ft/s? c. What is the height of the object at the highest point of its trajectory? d. With what speed does the object strike the ground?Beak length The length of the culmen (the upper ridge of a birds bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.941+4e1.65t, where L is measured in millimeters. a. Find L(1) and interpret the meaning of this value. b. Use a graph of L'(t) to describe how the culmen grows over the first 5 weeks of life.97REAntibiotic decay The half-life of an antibiotic in the bloodstream is 10 hours. If an initial dose of 20 milligrams is administered, the quantity left after t hours is modeled by Q(t)=20e0.0693t, for t 0. a. Find the instantaneous rate of change of the amount of antibiotic in the bloodstream, for t 0. b. How fast is the amount of antibiotic changing at t = 0? At t = 2? c. Evaluate and interpret limtQ(t) and limtQ(t).Population of the United States The population of the United States (in millions) by decade is given in the table, where t is the number of years alter 1910. These data are plotted and fitted with a smooth curve y=p(t) in the figure. a. Compute the average rate of population growth from 1950 to 1960. b. Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955. c. Estimate the instantaneous rate of growth in 1985.Growth rate of bacteria Suppose the following graph represents the number of bacteria in a culture t hours after the start of an experiment. a. At approximately what time is the instantaneous growth rate the greatest, for 0 t 36? Estimate the growth rate at this time. b. At approximately what time in the interval 0 t 36 is the instantaneous growth rate the least? Estimate the instantaneous growth rate at this time. c. What is the average growth rate over the interval 0 t 36?Velocity of a skydiver Assume the graph represents the distance (in m) fallen by a skydiver t seconds after jumping out of a plane. a. Estimate the velocity of the skydiver at t = 15. b. Estimate the velocity of the skydiver at t = 70. c. Estimate the average velocity of the skydiver between t = 20 and t = 90. d. Sketch a graph of the velocity function, for 0 t 120. e. What significant event occurred at t = 30?A function and its inverse function The function f(x)=xx+1 is one-to-one for x 1 and has an inverse on that interval. a. Graph f, for x 1. b. Find the inverse function f1 corresponding to the function graphed in part (a). Graph f1 on the same set of axes as in part (a). c. Evaluate the derivative of f1 at the point (12,1). d. Sketch the tangent lines on the graphs of f and f1 at (1,12) and (12,1). respectively.103RELimits The following limits represent the derivative of a function f at a point a. Find a possible f and a, and then evaluate the limit. 54. limh0sin2(4+h)12hLimits The following limits represent the derivative of a function f at a point a. Find a possible f and a, and then evaluate the limit. 55. limx5tan(3x11)x5Velocity of a probe A small probe is launched vertically from the ground. After it reaches its high point, a parachute deploys and the probe descends to Earth. The height of the probe above the ground is s(t)=300t50t2t3+2, for 0 t 6. a. Graph the height function and describe the motion of the probe. b. Find the velocity of the probe. c. Graph the velocity function and determine the approximate time at which the velocity is a maximum.107REMarginal and average cost Suppose a company produces fly rods. Assume C(x) = 0.0001x3 + 0.05x2 + 60x + 800 represents the cost of making x fly rods. a. Determine the average and marginal costs for x = 400 fly rods. b. Interpret the meaning of your results in part (a).Population growth Suppose p(t) = 1.7t3 + 72t2 + 7200t + 80,000 is the population of a city t years after 1950. a. Determine the average rate of growth of the city from 1950 to 2000. b. What was the rate of growth of the city in 1990?Position of a piston The distance between the head of a piston and the end of a cylindrical chamber is given by x(t)=8tt+1cm, for t 0 (measured in seconds). The radius of the cylinder is 4 cm. a. Find the volume of the chamber, for t 0. b. Find the rate of change of the volume V(f), for t 0. c. Graph the derivative of the volume function. On what intervals is the volume increasing? Decreasing?Boat rates Two boats leave a dock at the same time. One boat travels south at 30 mi/hr and the other travels east at 40 mi/hr. After half an hour, how fast is the distance between the boats increasing?Rate of inflation of a balloon A spherical balloon is inflated at a rate of 10 cm3/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?Rate of descent of a hot-air balloon A rope is attached to the bottom of a hot-air balloon that is floating above a flat field. If the angle of the rope to the ground remains 65 and the rope is pulled in at 5 ft/s, how quickly is the elevation of the balloon changing?Filling a tank Water flows into a conical tank at a rate of 2 ft3/min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.Angle of elevation A jet flies horizontally 500 ft directly above a spectator at an air show at 450 mi /hr. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?Viewing angle A man whose eye level is 6 ft above the ground walks toward a billboard at a rate of 2 ft/s. The bottom of the billboard is 10 ft above the ground, and it is 15 ft high. The mans viewing angle is the angle formed by the lines between the mans eyes and the top and bottom of the billboard. At what rate is the viewing angle changing when the man is 30 ft from the billboard?Shadow length A street light is fastened to the top of a 15-ft-high pole. If a 5-ft-tall women walks away from the pole in a straight line over level ground at a rate of 3 ft/s, how fast ts the length of her shadow changing when she is 10 ft away from the pole?Quadratic functions a. Show that if (a, f(a)) is any point on the graph of f(x) = x2, then the slope of the tangent line at that point is m = 2a. b. Show that if (a, f(a)) is any point on the graph of f(x) = bx2 + cx + d, then the slope of the tangent line at that point is m = 2ab + c.Derivative of the inverse in two ways Let f(x)=sinx,f1(x)=sin1(x), and (x0,y0)=(/4,1/2). a.Evaluate (f1)(1/2) using Theorem 3.21. b.Evaluate (f1)(1/2) directly by differentiating f1. Check for agreement with part (a).A parabola property Let f(x) = x2. a. Show that f(x)f(y)xy=f(x+y2), for all x y. b. Is this property true for f(x) = ax2, where a is a nonzero real number? c. Give a geometrical interpretation of this property. d. Is this property true for f(x) = ax3?Sketch the graph of a function that is continuous on an interval but does not have an absolute minimum value. Sketch the graph of a function that is defined on a closed interval but does not have an absolute minimum value.Consider the function f(x) = x3. Where is the critical point of f? Does f have a local maximum or minimum at the critical point?Sketch the graph of a function that is continuous on an open interval (a, b) but has neither an absolute maximum nor an absolute minimum value on (a, b).Sketch the graph of a function that has an absolute maximum, a local minimum, but no absolute minimum on [0, 3].What is a critical point of a function?Sketch the graph of a function f that has a local maximum value at a point c where f(c) = 0.Sketch the graph of a function f that has a local minimum value at a point c where f(c) is undefined.Absolute maximum/minimum values Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value. 11.Absolute maximum/minimum values Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value. 12.Absolute maximum/minimum values Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value. 13.Absolute maximum/minimum values Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value. 14.Local and absolute extreme values Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. 15.Local and absolute extreme values Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. 16.Local and absolute extreme values Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. 17.Local and absolute extreme values Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. 18.Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 23.f(x)=3x24x+2Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 24.f(x)=18x312xLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 25.f(x)=x339xLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 26.f(x)=x44x333x2+10Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 27.f(x)=3x3+3x222xLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 28.f(x)=4x553x3+5Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 29.f(x)=x34a2xLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 30.f(x)=x5tan1xLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 31.f(t)=tt2+1Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 32.f(x)=12x520x3Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 33.f(x)=ex+ex2Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 34.f(x)=sinxcosxLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 35.f(x)=1x+lnxLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 36.f(t)=t22ln(t2+1)Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 37.f(x)=x2x+5Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 38.f(x)=(sin1x)(cos1x)Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 39.f(x)=xxaLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 40.f(x)=xxaLocating critical points Find the critical points of the following functions. Assume a is a nonzero constant. 41.f(t)=15t5a4t42EAbsolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 43.f(x)=x210on[2,3]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 44.f(x)=(x+1)4/3on[9,7]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 45.f(x)=x33x2on[1,3]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 46.f(x)=x44x3+4x2on[1,3]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 47.f(x)=3x525x3+60xon[2,3]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 48.f(x)=2exx2on[0,2e]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 49.f(x)=cos2xon[0,]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 50.f(x)=x(x2+3)2on[2,2]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 51.f(x)=sin3xon[/4,/3]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 52.f(x)=3x2/3xon[0,27]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 53.f(x)=(2x)xon[0.1,1]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 54.f(x)=xe1x/2on[0,5]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 55.f(x)=x2+cos1xon[1,1]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 56.f(x)=x2x2on[2,2]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 57.f(x)=2x315x2+24xon[0,5]58EAbsolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 59.f(x)=4x33+5x26xon[4,1]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 60.f(x)=2x615x4+24x2on[2,2]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 61.f(x)=x(x2+9)5on[2,2]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 62.f(x)=x1/2(x254)on[0,4]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 63.f(x)=secxon[4,4]64EAbsolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 65.f(x)=x3exon[1,5]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 66.f(x)=xlnx5on[0.1,5]Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 67.f(x)=x2/3(4x2)on[2,2]68EEfficiency of wind turbines A wind Turbine converts wind energy into electrical power. Let v1 equal the upstream velocity of the wind before it encounters the wind turbine, and let v2 equal the downstream velocity of the wind after it passes through the area swept out by the turbine blades. a.Assuming that v1 0, give a physical explanation to show that 0v2v11. b.The amount of power extracted from the wind depends on the ratior=v2v1, the ratio of the downstream velocity to upstream velocity. Let R(r) equal the fraction of power that is extracted from the total available power in the wind stream, for a given value of r. In about 1920, the German physicist Albert Betz showed that R(r)=12(1+r)(1r2), where 0r1 (a derivation of R is outlined in Exercise 70). Calculate R(1) and explain how you could have arrived at this value without using the formula for R. Give a physical explanation of why it is unlikely or impossible for it to be the case that r = 1. c.Calculate R(0) and give a physical explanation of why it is unlikely or impossible for it to be the case that r = 0. d.The maximum value of R is called the Betz limit. It represents the theoretical maximum amount of power that can be extracted from the wind. Find this value and explain its physical meaning.Derivation of wind turbine formula A derivation of toe function R in Exercise 69, based on three equations from physics, is outlined here. Consider again the figure given in Exercise 69, where v_1 equals the upstream velocity of the wind just before the wind stream encounters the wind turbine, and equals the downstream velocity of the wind just after the wind stream passes through the area swept out by the turbine blades. An equation for the power extracted by the rotor blades, based on conservation of momentum, is P=v2A(v1v2), where v is the velocity of the wind (in m/s) as it passes through the turbine blades, p is the density of air (in kg/m3), and A is the area (in m2) of the circular region swept out by the rotor blades. a.Another expression for the power extracted by the rotor blades, based or conservation of energy is P=12vA(v12v22). Equate the two power equations and solve for v. b.Show that P=A4vA(v1+v2)(v12v22). c.If the wind were to pass through the same area A without being disturbed by rotor blades, the amount of available power would be P0=Av132. Let r=v2v1 and simplify the ratio PP0 to obtain the function R(r) given in Exercise 69.Suppose the position of an object moving horizontally after t seconds is given by the function s(t)=32tt4, where 0t3 and s is measured in feet, with s 0 corresponding to positions to the right of the origin. When is the object farthest to the right?Minimum surface area box All boxes with a square base and a volume of 50 ft3 have a surface area given by S(x) = 2x2 + 200/x, where x is the length of the sides of the base. Find the absolute minimum of the surface area function on the interval (0, ). What are the dimensions of the box with minimum surface area?Trajectory high point A stone is launched vertically upward from a cliff 192 feet above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = 16t2 + 64t + 192, for 0 t 6. When does the stone reach its maximum height?Maximizing revenue A sales analyst determines that the revenue from sales of fruit smoothies is given by R(x) = 60x2 + 300x, where x is the price in dollars charged per item, for 0 x 5. a. Find the critical points of the revenue function. b. Determine the absolute maximum value of the revenue function and the price that maximizes the revenue.Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking n people on a city tour is P(n) = n(50 0.5n) 100. (Although P is defined only for positive integers, treat it as a continuous function.) a. How many people should the guide take on a tour to maximize the profit? b. Suppose the bus holds a maximum of 45 people. How many people should be taken on a tour to maximize the profit?Maximizing rectangle perimeters All rectangles with an area of 64 have a perimeter given by P(x) = 2x + 128/x, where x is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function on the interval (0, ). What are the dimensions of the rectangle with minimum perimeter?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function f(x)=x has a local maximum on the interval [0, ). b. If a function has an absolute maximum on a closed interval, then the function must be continuous on that interval. c. A function f has the property that f(2) = 0. Therefore, f has a local extreme value at x = 2. d. Absolute extreme values of a function on a closed interval always occur at a critical point or an endpoint of the interval.Absolute maxima and minima a. Find the critical points of f on the given interval. b. Determine the absolute extreme values of f on the given interval. c. Use a graphing utility to confirm your conclusions. 63. f(x)=x/x4 on [6, 12]Absolute maxima and minima a. Find the critical points of f on the given interval. b. Determine the absolute extreme values of f on the given interval. c. Use a graphing utility to confirm your conclusions. 57. f(x) = 2x sin x on [2, 6]Critical points and extreme values a. Find the critical points of the following functions on the given interval. b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither. c. Find the absolute maximum and minimum values on the given interval when they exist. 68. f(x) = 6x4 16x3 45x2 + 54x + 23 on [5, 5]Critical points and extreme values a. Find the critical points of the following functions on the given interval. b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither. c. Find the absolute maximum and minimum values on the given interval when they exist. 69. f() = 2 sin + cos on [2, 2]82E83EAbsolute value functions Graph the following functions and determine the local and absolute extreme values on the given interval. 74. f(x) = |x 3| + |x + 2| on [4, 4]85E86EEvery second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 m from a point Q on the shore that is 50 m from you (see figure). If you can swim at a speed of 2 m/s and run at a speed of 4 m/s, at what point along the shore, x meters from Q, should you stop running and start swimming if you want to reach the swimmer in the minimum time? a. Find the function T that gives the travel time as a function of x, where 0 x 50. b. Find the critical point of T on (0, 50). c. Evaluate T at the critical point and the endpoints (x = 0 and x = 50) to verify that the critical point corresponds to an absolute minimum. What is the minimum travel time? d. Graph the function T to check your work.Extreme values of parabolas Consider the function f(x) = ax2 + bx + c, with a 0. Explain geometrically why f has exactly one absolute extreme value on (, ). Find the critical point to determine the value of x at which f has an extreme value.Values of related functions Suppose f is differentiable on (, ) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = x f(x) + 1 and let h(x) = x f(x) + x + 1, for all values of x. a. Evaluate g(2), h(2), g(2), and h(2). b. Does either g or h have a local extreme value at x = 2? Explain.90EProof of the Local Extreme Value Theorem Prove Theorem 4.2 for a local maximum: If f has a local maximum value at the point c and f(c) exists, then f(c) = 0. Use the following steps. a. Suppose f has a local maximum at c. What is the sign of f(x) f(c) if x is near c and x c? What is the sign of f(x) f(c) if x is near c and x c? b. If f(c) exists, then it is defined by limxcf(x)f(c)xc. Examine this limit as x c + and conclude that f(c) 0. c. Examine the limit in part (b) as x c and conclude that f(c) 0. d. Combine parts (b) and (c) to conclude that f(c) = 0.92EWhere on the interval [0, 4] does f(x) = 4x x2 have a horizontal tangent line?Sketch the graph of a function that illustrates why the continuity condition of the Mean Value Theorem is needed. Sketch the graph of a function that illustrates why the differentiability condition of the Mean Value Theorem is needed.Give two distinct linear functions f and g that satisfyf(x)=g(x); that is, the lines have equal slopes. Show that f and g differ by a constant.Explain Rolles Theorem with a sketch.Draw the graph of a function for which the conclusion of Rolles Theorem does not hold.Explain why Rolles Theorem cannot be applied to the function f(x) =|x| on the interval [a, a], for any a 0.Explain the Mean Value Theorem with a sketch.For each function f and interval [a, b], a graph of f is given along with the secant line that passes though the graph of f at x = a and x = b. a.Use the graph to make a conjecture about the value(s) of c satisfying the equation f(b)f(a)ba=f(c). b.Verify your answer to part (a) by solving the equation f(b)f(a)ba=f(c) for c. 5.f(x)=x24+1;[2,4]For each function f and interval [a, b], a graph of f is given along with the secant line that passes though the graph of f at x = a and x = b. a.Use the graph to make a conjecture about the value(s) of c satisfying the equation f(b)f(a)ba=f(c). b.Verify your answer to part (a) by solving the equation f(b)f(a)ba=f(c) for c. 6.f(x)=2x;[0,4]For each function f and interval [a, b], a graph of f is given along with the secant line that passes though the graph of f at x = a and x = b. a.Use the graph to make a conjecture about the value(s) of c satisfying the equation f(b)f(a)ba=f(c). b.Verify your answer to part (a) by solving the equation f(b)f(a)ba=f(c) for c. 7.f(x)=x516;[2,2]At what points c does the conclusion of the Mean Value Theorem hold for f(x) = x3 on the interval f [10, 10]?Draw the graph of a function for which the conclusion of the Mean Value Theorem does not hold.