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All Textbook Solutions for Calculus Volume 1

For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line. 315. [T] xy+sin(x)=1,(2,0)[T] The graph of a folium of Descartes with equation 2x3+2y39xy=0 is given in the following graph. Find the equation of the tangent line at the point (2, 1). Graph the tangent line along with the folium. Find the equation of the normal line to the tangent line in a. at the point (2, 1).For the equation x2+2xy3y2=0 , Find the equation of the normal to the tangent line at the point (1, 1). At what other point does the normal line in a. intersect the graph of the equation?Find all points on the graph of y327y=x290 at which the tangent line is vertical.For the equation x2+xy+y2=7 , Find the x -intercept(s). Find the slope of the tangent line(s) at the x-intercept(s). What does the value(s) in b. indicate about the tangent line(s)?Find the equation of the tangent line to the graph of the equation sin1x+sin1y=6 at the point (0,12) .Find the equation of the tangent line to the graph ofthe equation tan1(x+y)=x2+4 at the point (0,1).Find y and y for x2+6xy2y2=3 .T] The number of cell phones produced when xdollars isspent on labor and ydollars is spent on capital invested by a manufacturer can be modeled by the equation 60x3/4y1/4=3240 . Find dydx and evaluate at the point (81,16). Interpret the result of a.T] The number of cars produced when x dollars is spent on labor and y dollars is spent on capital invested by a manufacturer can be modeled by the equation 30x1/3y2/3=360 . (Both x and y are measured inthousands of dollars.) Find dydx and evaluate at the point (27, 8). Interpret the result of a.The volume of a right circular cone of radius xand height y is given by V=13x2y . Suppose that thevolume of the cone is 85cm3 . Find dydx when x = 4 and y = 16 For the following exercises, consider a closed rectangular box with a square base with side x and height y. 326. Find an equation for the surface area of the rectangular box, S(x, y).For the following exercises, consider a closed rectangular box with a square base with side x and height y. 327. If the surface area of the rectangular box is 78 square feet, find dydx when x = 3 feet and y= 5 feet.For the following exercises, use implicit differentiation to determine y . Does the answer agree with the formulas we have previously determined? 328. x=siny For the following exercises, use implicit differentiation to determine y . Does the answer agree with the formulas we have previously determined? 329. x=cosy For the following exercises, use implicit differentiation to determine y . Does the answer agree with the formulas we have previously determined? 330. x=tany For the following exercises, find f(x) for each function. 331. f(x)=x2ex For the following exercises, find f(x) for each function. 332. f(x)=exx .For the following exercises, find f(x) for each function. 333. f(x)=ex3Inx For the following exercises, find f(x) for each function. 334. f(x)=e2x+2x For the following exercises, find f(x) for each function. 335. f(x)=exexex+ex For the following exercises, find f (x) for each function.For the following exercises, find f(x) for each function. 337. f(x)=24x+4x2 For the following exercises, find f(x) for each function. 338. f(x)=3sin3x For the following exercises, find f(x) for each function. 339. f(x)=xx For the following exercises, find f(x) for each function. 340. f(x)=In(4x3+x)For the following exercises, find f(x) for each function. 341. f(x)=In5x7 For the following exercises, find f(x) for each function. 342. f(x)=x2In9x For the following exercises, find f(x) for each function. 343. f(x)=log(sec)x For the following exercises, find f(x) for each function. 344. f(x)=log7(6x4+3)5 For the following exercises, find f(x) for each function. 345. f(x)=2xlog37x24 For the following exercises, use logarithmic differentiation to find dydx . 346. y=xx For the following exercises, use logarithmic differentiation to find dydx . 347. y=(sin2x)4x For the following exercises, use logarithmic differentiation to find dydx . 348. y=(Inx)Inx For the following exercises, use logarithmic differentiation to find dydx . 349. y=xlog2x For the following exercises, use logarithmic differentiation to find dydx . 350. y=(x21)Inx For the following exercises, use logarithmic differentiation to find dydx . 351. y=xcotx For the following exercises, use logarithmic differentiation to find dydx . 352. y=x+11x243 For the following exercises, use logarithmic differentiation to find dydx . 353. y=x1/2(x2+3)2/3(3x4)4 [T] Find an equation of the tangent line to the graph of f(x)=4xe(x21) at the point where x = -1. Graph both the function and the tangent line.[T] Find the equation of the line that is normal to the graph of f(x)=x5x . at the point where x = 1. Graph both the function and the normal line.[T] Find the equation of the tangent line to the graph of x3xIny+y3=2x+5 at the point where x = 2. (Hint: Use implicit differentiation to find dydx .) Graph both the curve and the tangent line.Consider the function y=x1/x for x > 0. Determine the points on the graph where the tangent line is horizontal. Determine the points on the graph where y' > 0 and those where y' < 0 The formula I(t)=sintet is the formula for a decaying alternating current. a. Complete the following table with the appropriate values. t sintet 0 (i) 2 (ii) (iii) 32 (iv) 2 (v) 2 (vi) 3 (vii) 72 (viii) 4 (ix) b. Using only the values in the table, determine where the tangent line to the graph of I(t) is horizontal.[T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year. Write the exponential function that relates the total population as a function of t. Use a. to determine the rate at which the population is increasing in t years. Use b. to determine the rate at which the population is increasing in 10 years.[T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present. Write the exponential function that relates the amount of substance remaining as a function of t, measured in hours. Use a. to determine the rate at which the substance is decaying in t hours. Use b. to determine the rate of decay at t = 4 hours.[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the functionN(t)=5.3e0.093t20.87t,(0t4) , where N(t) gives the number of cases (in thousands) and t is measured in years, with t = 0 corresponding to the beginning of 1960. Show work that evaluates N(0) and N(4). Briefly describe what these values indicate about the disease in New York City. Show work that evaluates N’(0) and N’(3). Briefly describe what these values indicate about the disease in the United States.[T] The relative rate of change of a differentiable function y=f(x) is given by . 100f(x)f(x) . One model for population growth is a Gompertz growth function, given by P(x)=aebecx where a, b, and c are constants. Find the relative rate of change formula for the generic Gompertz function. Use a. to find die relative rate of change of a population in x = 20 months when a = 204. b = 0.0198. and c = 0.15. Briefly interpret what the result of b. means.For the following exercises, use the population of New York City from 1790 to 1860, given in the following table. Years since 1790 Population 0 33,131 10 60,515 20 96,373 30 123,706 40 202,300 50 312,710 60 515,547 70 813,669 Table 3.9 New York City Population Over Time Source: http://en.wikipedia.org/ wiki/ Largest_cities _in_the_United_States _by_population_by_decade. 363. [T] Using a computer program or a calculator, fit a growth curve to the data of the form p=abt .For the following exercises, use the population of New York City from 1790 to 1860, given in the following table. Years since 1790 Population 0 33,131 10 60,515 20 96,373 30 123,706 40 202,300 50 312,710 60 515,547 70 813,669 Table 3.9 New York City Population Over Time Source: http://en.wikipedia.org/ wiki/ Largest_cities _in_the_United_States _by_population_by_decade. 364. [T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.For the following exercises, use the population of New York City from 1790 to 1860, given in the following table. Years since 1790 Population 0 33,131 10 60,515 20 96,373 30 123,706 40 202,300 50 312,710 60 515,547 70 813,669 Table 3.9 New York City Population Over Time Source: http://en.wikipedia.org/ wiki/ Largest_cities _in_the_United_States _by_population_by_decade. 365. [T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.For the following exercises, use the population of New York City from 1790 to 1860, given in the following table. Years since 1790 Population 0 33,131 10 60,515 20 96,373 30 123,706 40 202,300 50 312,710 60 515,547 70 813,669 Table 3.9 New York City Population Over Time Source: http://en.wikipedia.org/ wiki/ Largest_cities _in_the_United_States _by_population_by_decade. 366. [T] Using the tables of first and second derivatives and the best fit, answer the following questions: Will the model be accurate in predicting the future population of New York City? Why or why not? Estimate the population in 2010. Was the prediction collect from a.?True or False? Justify the answer with a proof or a counterexample. 367. Every function has a derivative.True or False? Justify the answer with a proof or a counterexample. 368. A continuous function has a continuous derivative.True or False? Justify the answer with a proof or a counterexample. 369. A continuous function has a derivative.True or False? Justify the answer with a proof or a counterexample. 370. If a function is differentiable, it Is continuous.Use the Limit definition of the derivative to exactly evaluate the derivative. 371. f(x)=x+4 Use the Limit definition of the derivative to exactly evaluate the derivative. 372. f(x)=3x Find the derivatives of the following functions. 373. f(x)=3x34x2 Find the derivatives of the following functions. 374. f(x)=(4x2)3 Find the derivatives of the following functions. 375. f(x)=esinx Find the derivatives of the following functions. 376. f(x)=In(x+2)Find the derivatives of the following functions. 377. f(x)=x2cox+xtan(x)Find the derivatives of the following functions. 378. f(x)=3x2+2 Find the derivatives of the following functions 379. 379. f(x)=x4sin1(x)Find the derivatives of the following functions. 380. x2y=(y+2)+xysin(x)Find the following derivatives of various orders 381. 381. First derivative of y=xIn(x)cosx Find the following derivatives of various orders 382. Third derivative of y=(3x+2)2 Find the following derivatives of various orders. 383. Second derivative of y=4x+x2sin(x)Find the equation of the tangent line to the following equations at the specified point. 384. y=cos1(x)+x at x=0 Find the equation of the tangent line to the following equations at the specified point. 385. y=x+ex1x at x=1 Draw the derivative for the following graphs. 386.Draw the derivative for the following graphs. 387. The following questions concern the water level in Ocean City, New Jersey, in january, which can be approximated by w(t)=1.9+2.9cos(6t) , where t is measured in hours after midnight, and the height is measured in feet.Find and graph the derivative. What is the physical meaning?Find w' (3). What is the physical meaning of this value? The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table. Hours after midnight August26 Wind speed (mph) 1 45 5 75 11 100 29 115 49 145 58 175 73 155 81 125 58 95 107 35 Table 3.10 Wind Speeds of Hurricane Katrina Source: http://news.nationalgeograptiic.com/news/2005/ 09/0914_050914_katrina_timeline.html.Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning? Hours after midnight August 26 Wind speed (mph) 1 45 5 75 11 100 29 115 49 145 58 175 73 155 81 125 58 95 107 35 Table 3.10 Wind Speeds of Hurricane Katrina Source: http://news.nationalgeograptiic.com/news/2005/ 09/0914_050914_katrina_timeline.html Estimate the derivative of the wind speed at hour 83. What is the physical meaning?For the following exercises, find the quantities for the given equation. 1. Find dydt at x=1 and y=x2+3 if dxdt=4 .For the following exercises, find the quantities for the given equation. 2. Find dxdt at x=2 and y=2x2+1 if dydt=1 .For the following exercises, find the quantities for the given equation. Find dzdt at (x, y) = (1, 3) and z2=x2+y2 if dxdt=4 and dydt=3.For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities. 4. [T] If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, Q) is given by the equation 1R=1R1+1R2 . If R1 is increasing at a rate of 0.5/min and R2 decreases at a rate of 1.1/min , at what rate does the total resistance change when R1=20 and R2=50/min ?For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities. 5. A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?A 25-ft ladder is leaning against a wall. If we push the ladder toward the wall at a rate of 1 ft/sec, and the bottom of the ladder is initially 20 ft away from the wall, how fast does the ladder move up the wall 5 sec after we start pushing?Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at 300 mi/h. If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B, at what rate is the distance between the airplanes changing?You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding 12 mph north. After you traveled 4 mi, at what rate is the distance between you changing?Two buses are driving along parallel freeways that are 5 mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55 mph, find the rate at which the distance between the buses is changing when they are 13 mi apart, heading toward each other.A 6-ft-tall person walks away from a 10-ft lamppost at a constant rate of 3 ft/sec. What is the rate that the tip of the shadow moves away from the pole when the person is 10 ft away from the pole?Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is 10 ft from the pole?A 5-ft-tall person walks toward a wall at a rate of 2 ft/ sec. A spotlight is located on the ground 40 ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is 10 ft from the wall?Using the previous problem, what is the rate at which the shadow changes when the person is 10 ft from the wall, if the person is walking away from the wall at a rate of 2 ft/ sec?A helicopter starting on the ground is rising directly into the air at a rate of 25 ft/sec. You are running on the ground starting directly tinder the helicopter at a rate of 10 ft/sec. Find the rate of change of the distance between the helicopter and yourself after 5 sec.Using the previous problem, what is the rate at which the distance between you and the helicopter is changing when the helicopter has risen to a height of 60 ft in the air, assuming that, initially, it was 30 ft above you?For the following exercises, draw and label diagrams to help solve the related-rates problems. 16. The side of a cube increases at a rate of 12m/sec . Find the rate at which the volume of the cube increases when the side of the cube is 4 m.For the following exercises, draw and label diagrams to help solve the related-rates problems. 17. The volume of a cube decreases at a rate of 10 m/sec. Find the rate at which the side of the cube changes when the side of the cube is 2 m.For the following exercises, draw and label diagrams to help solve the related-rates problems. 18. The radius of a circle increases at a rate of 2 m/sec. Find the rate at which the area of the circle increases when the radius is 5 m.For the following exercises, draw and label diagrams to help solve the related-rates problems. 19. The radius of a sphere decreases at a rate of 3 m/sec. Find the rate at which the surface area decreases when the radius is 10 m.For the following exercises, draw and label diagrams to help solve the related-rates problems. 20. The radius of a sphere increases at a rate of 1 m/sec. Find the rate at which the volume increases when the radius is 20 m.For the following exercises, draw and label diagrams to help solve the related-rates problems. 21. The radius of a sphere is increasing at a rate of 9 cm/ sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.For the following exercises, draw and label diagrams to help solve the related-rates problems. 22. The base of a triangle is shrinking at a rate of 1 cm/min and the height of the triangle is increasing at a rate of 5 cm/min. Find the rate at which the area of the triangle changes when the height is 22 cm and the base is 10 cm.For the following exercises, draw and label diagrams to help solve the related-rates problems. 23. A triangle has two constant sides of length 3 ft and 5 ft. The angle between these two sides is increasing at a rate of 0.1 rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is /6 .For the following exercises, draw and label diagrams to help solve the related-rates problems. 24. A triangle has a height that is increasing at a rate of 2 cm/sec and its area is increasing at a rate of 4 cm2/sec. Find the rate at which the base of the triangle is changing when the height of the triangle is 4 cm and the area is 20 cm2.For the following exercises, consider a tight cone that is leaking water. The dimensions of die conical tank are a height of 16 ft and a radius of 5 ft. 25. How fast does the depth of the water change when the water is 10 ft high if the cone leaks water at a rate of 10 ft3/min?For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. 26. Find the rate at which the surface area of the water changes when the water is 10 ft high if the cone leaks water at a rate of 10 ft3/min.For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. 27. If the water level is decreasing at a rate of 3 in./min when the depth of the water is 8 ft, determine the rate at which water is leaking out of the cone.For the following exercises, consider a tight cone that is leaking water. The dimensions of die conical tank are a height of 16 ft and a radius of 5 ft. 28. A vertical cylinder is leaking water at a rate of 1 ft3/sec. If the cylinder has a height of 10 ft and a radius of 1 ft, at what rate is the height of the water changing when the height is 6 ft?For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. 29. A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of 2 m and a radius of 2 m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is 10 cm/min when the height is 1 m.A trough has ends shaped like isosceles triangles, with width 3 m and height 4 m, and the trough is 10 m long. Water is being pumped into the trough at a rate of 5m3/min. At what rate does the height of the water change when the water is 1 m deep?A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of 23m/sec ?For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft3/min. The radius of the pool is 10 ft. 32. Find the rate at which the depth of the water is changing when the water has a depth of 5 ft.For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft3/min. The radius of the pool is 10 ft. 33. Find the rate at which the depth of the water is changing when the water has a depth of 1 ft.For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft3/min. The radius of the pool is 10 ft. 34. If the height is increasing at a rate of 1 in./sec when the depth of the water is 2 ft, find the rate at which water is being pumped in.For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft3/min. The radius of the pool is 10 ft. 35. Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 ft3/min. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 ft.For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft3/min. The radius of the pool is 10 ft. 36. Using a similar setup from the preceding problem, find the rate at which die gravel is being unloaded if the pile is 5 ft high and the height is increasing at a rate of 4 in./min.For the following exercises, draw the situations and solve the related-rate problems. 37. You are stationary on the ground and are watching a bird fly horizontally at a rate of 10 m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?For the following exercises, draw the situations and solve the related-rate problems. 38. You stand 40 ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/ sec. Find the rate at which the angle of elevation changes when the rocket is 30 ft in the air.A lighthouse, L, is on an island 4 mi away from the closest point, P, on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach 1 mi away from the closest point on the beach.You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus navels west at a rate of 10 m/ sec away from the intersection - you have missed die bus! What is die rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection? For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.[T] A batter hits a ball toward third base at 75 ft/sec and runs toward first base at a rate of 24 ft/sec. At what rate does the distance between the ball and the batter change when 2 sec have passed?[T] A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)[T] A batter hits the ball and runs toward first base at a speed of 22 ft/sec. At what rate does the distance between the runner and second base change when the runner has run 30 ft?Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of 18 ft/sec toward second base and the runner at second base runs at a speed of 20 ft/sec toward third base. How fast is the distance between runners changing 1 sec after the ball is hit?What is the linear approximation for any generic linear function y=mx+b ?Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.Explain why the linear approximation becomes less accurate as you increase the distance between x and a. Use a graph to prove your argument.When is the linear approximation exact?For the following exercises, find the linear approximation L(x) to y = f(x) near x = a for the function. 50. [T] f(x)=x+x4,a=0 ]For the following exercises, find the linear approximation L(x) to y = f(x) near x = a for the function. 51. [T] f(x)=1x,a=2 For the following exercises, find the linear approximation L(x) to y = f(x) near x = a for the function. 52. [T] f(x)=tanx,a=4 For the following exercises, find the linear approximation L(x) to y = f(x) near x = a for the function. 53. [T] f(x)=sinx,a=2 For the following exercises, find he linear approximation L(x) to y = f(x) near x = a for the function. 54. [T] f(x)=xsinx,a=2 For the following exercises, find the linear approximation L(x) to y = f(x) near x = a for the function. 55. [T] f(x)=sin2x,a=0 For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 56. [T] (2.0001)6 For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 57. [T] sin(0.02)For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a, and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 58. [T] cos(0.03)For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a, and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 59. [T] (15.99)1/4 For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a, and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 60. [T] 10.98 For the following exercises, compute the values given within 0.01 by deciding on the appropriate f(x) and a, and evaluating L(x)=f(a)+f(a)(xa). Check your answer using a calculator. 61. [T] sin(3.14)For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 62. (1.01)3 For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 63. cos(0.01)For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 64. (sin( 0.01))2 For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 65. (1.01)3 For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 66. (1+1 10)10 For the following exercises, determine the appropriate f(x) and a, and evaluate L(x)=f(a)+f(a)(xa). Calculate the numerical error in the linear approximations that follow. 67. 8.99 For the following exercises, find the differential of the function. 68. y=3x4+x22x+1 For the following exercises, find the differential of the function. 69. y=xcosx For the following exercises, find the differential of the function. 70. y=1+x For the following exercises, find the differential of the function. 71. y=x2+2x1 For the following exercises, find the differential and evaluate for the given x and dx. 72. y=3x2x+6,x=2,dx=0.1 For the following exercises, find the differential and evaluate for the given x and dx. 73. y=1x+1,x=1,dx=0.25 For the following exercises, find the differential and evaluate for the given x and dx. 74. y=tanx,x=0,dx=10 For the following exercises, find the differential and evaluate for the given x and dx. 75. y=3x2+2x+1,x=0,dx=0.1 For the following exercises, find the differential and evaluate for the given x and dx. 76. y=sin(2x)x,x=,dx=0.25 For the following exercises, find the differential and evaluate for the given x and dx. 77. y=x3+2x+1x,x=1,dx=0.05 For the following exercises, find the change in volume dV or in surface area dA. 78. dV if the sides of a cube change from 10 to 10.1.For the following exercises, find the change in volume dV or in surface area dA. 79. dA if the sides of a cube change from x to x+dx .For the following exercises, find the change in volume dV or in surface area dA. 80. dA if the radius of a sphere changes from r by dr.For the following exercises, find the change in volume dV or in surface area dA. 81. dV if the radius of a sphere changes from r by dr.For the following exercises, find the change in volume dV or in surface area dA. 82. dV if a circular cylinder with r = 2 changes height from 3 cm to 3.05 cm.For the following exercises, find the change in volume dV or in surface area dA. 83. dV if a circular cylinder of height 3 changes from r = 2 to r = 1.9 cm.For the following exercises, use differentials to estimate the maximum and relative error when computing the surface area or volume. 84. A spherical golf ball is measured to have a radius of 5 mm, with a possible measurement error of 0.1 mm. What is the possible change in volume?For the following exercises, use differentials to estimate the maximum and relative error when computing the surface area or volume. 85. A pool has a rectangular base of 10 ft by 20 ft and a depth of 6 ft. What is the change in volume if you only fill it up to 5.5 ft?For the following exercises, use differentials to estimate the maximum and relative error when computing the surface area or volume. 86. An ice cream cone has height 4 in. and radius 1 in. If the cone is 0.1 in. thick, what is the difference between the volume of the cone, including the shell, and the volume of the ice cream you can fit inside the shell?For the following exercises, confirm the approximations by using the linear approximation at x = 0. 87. 1x112x For the following exercises, confirm the approximations by using the linear approximation at x = 0. 88. 11x21 For the following exercises, confirm the approximations by using the linear approximation at x = 0. 89. c2+x2c In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation y=ax2+bx+c , which was m=b(2a) . Prove this formula using calculus.If you are finding an absolute minimum over an interval [a, b], why do you need to check the endpoints? Draw a graph that supports your hypothesis.If you are examining a function over an interval (a, b), for a and b finite, is it possible not to have an absolute maximum or absolute minimum?When you are checking for critical points, explain why you also need to determine points where f(x) is undefined. Draw a graph to support your explanation.Can you have a finite absolute maximum for y=ax2+bx+c over (,) ? Explain why or why not using graphical arguments.Can you have a finite absolute maximum for y=ax3+bx2+cx+d over (,) assuming a is non zero? Explain why or why not using graphical arguments.Let m be the number of local minima and M be the number of local maxima. Can you create a function where Mm+2 ? Draw a graph to support your explanation.Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible.[T] Graph the function y=eax . For which values of a, on any infinite domain, will you have an absolute minimum and absolute maximum?For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. 100.For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. 101.For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. 102.For the following exercises, determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. 103.For the following problems, draw graphs of f(x), which is continuous, over the interval [-4, 4] with the following properties: 104. Absolute maximum at x = 2 and absolute minima at x=3 For the following problems, draw graphs of f(x), which is continuous, over the interval [-4, 4] with the following properties: 105. Absolute minimum at x = 1 and absolute maximum at x = 2 For the following problems, draw graphs of f(x), which is continuous, over the interval [-4, 4] with the following properties: 106. Absolute maximum at x = 4, absolute minimum at x = -1, local maximum at x = -2, and a critical point that is not a maximum or minimum at x = 2 For the following problems, draw graphs of f(x), which is continuous, over the interval [-4, 4] with the following properties: 107. Absolute maxima at x = 2 and x = -3, local minimum at x = 1, and absolute minimum at x = 4 For the following exercises, find the critical points in the domains of the following functions. 108. y=4x33x For the following exercises, find the critical points in the domains of the following functions. 109. y=4xx2 For the following exercises, find the critical points in the domains of the following functions. 110. y=1x1 For the following exercises, find the critical points in the domains of the following functions. 111. y=In(x2)For the following exercises, find the critical points in the domains of the following functions. 112. y=tan(x)For the following exercises, find the critical points in the domains of the following functions. 113. y=4x2 For the following exercises, find the critical points in the domains of the following functions. 114. y=x3/23x5/2 For the following exercises, find the critical points in the domains of the following functions. 115. y=x21x2+2x3 For the following exercises, find the critical points in the domains of the following functions. 116. y=sin2(x)For the following exercises, find the critical points in the domains of the following functions. 117. y=x+1x For the following exercises, find the local and or absolute maxima for the functions over the specified domain. 118. f(x)=x2+3 over [-1, 4]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 119. y=x2+2x over [1, 4]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 120. y=(xx2)2 over [-1, 1]For the following exercises, find the local and or absolute maxima for the functions over the specified domain. 121. y=1(xx2) over [0, 1]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 122. y=9x over [1, 9]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 123. y=x+sin(x) over [0, 2 ]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 124. y=x1+x over [0, 100]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 125. y=|x+1|+|x1| over [-3, 2]For the following exercises, find the local and/or absolute maxima for the functions over the specified domain. 126. y=xx3 over [0, 4]For the following exercises, find the local and or absolute maxima for the functions over the specified domain. 127. y=sinx+cosx over [0, 2 ]For the following exercises, find the local and or absolute maxima for the functions over the specified domain. 128. y=4sin3cos over [0, 2 ]For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 129. y=x2+4x+5 For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 130. y=x312x For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 131. y=3x4+8x318x2 For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 132. y=x3(1x6)For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 133. y=x2+x+6x1 For the following exercises, find the local and absolute minima and maxima for the functions over (,) . 134. y=x21x1 For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 135. [T] y=3x1x2 For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 136. [T] y=x+sin(x)For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 137. [T] y=12x5+45x4+20x390x2120x+3 For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 138. [T] y=x3+6x2x30x2 For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly. 139. [T] y=4x24+x2 A company that produces cell phones has a cost function of C=x21200x+36,400 , where C is cost in dollars and x is number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes this cost function?A ball is thrown into the air and its position is given by h(t)=4.9t2+60t+5m . Find the height at which the ball stops ascending. How long after it is thrown does this happen?For the following exercises, consider the production of gold during the California gold rush (1848-1888). The production of gold can be modeled by G(t)=(25t)(t2+16) , where t is the number of years since the rush began (0t40) and G is ounces of gold produced (in millions). A summary of the data is shown in the following figure. 142. Find when the maximum (local and global) gold production occurred, and the amount of gold produced during that maximum.For the following exercises, consider the production of gold during the California gold rush (1848-1888). The production of gold can be modeled by G(t)=(25t)(t2+16) , where t is the number of years since the rush began (0t40) and G is ounces of gold produced (in millions). A summary of the data is shown in the following figure. 143. Find when the minimum (local and global) gold production occurred. What was the amount of gold produced during this minimum?Find the critical points, maxima, and minima for the following piecewise functions. 144. y={x24x0x1x241x2 Find the critical points, maxima, and minima for the following piecewise functions. y={x2+1x1x24x+5x1 .For the following exercises, find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions. 146. y=ax2+bx+c , given that a > 0 For the following exercises, find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions. 147. y=(x1)a , given that a >1 Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.When are Rolle's theorem and the Mean Value Theorem equivalent?If you have a function with a discontinuity, is it still possible to have f(c)(ba)=f(b)f(a) ? Draw such an example or prove why not.For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 152. y=sin(x)For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 153. y=1x3 For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 154. y=4x2 For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 155. y=x24 For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 156. y=In(3x5)For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f(c)(ba)=f(b)f(a) 157. [T] y=3x3+2x+1 over [-1, 1]For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f(c)(ba)=f(b)f(a) 158. [T] y=tan(4x)over[32,32]For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f(c)(ba)=f(b)f(a) 159. [T] y=x2cos(x) over [-2, 2]For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f(c)(ba)=f(b)f(a) 160. [T] y=x634x598x4+1516x3+332x2+316x+132 over [-1, 1]For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 161. f(x)=x3 For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 162. f(x)=sin(x)For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 163. f(x)=cos(2x)For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 164. f(x)=1+x+x2 For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 165. f(x)=(x1)10 For the following exercises, use the Mean Value Theorem and find all points 0 < c < 2 such that f(2)f(0)=f(c)(20) . 166. f(x)=(x1)9 For the following exercises, show there is no c such that f(1)f(1)=f(c)(2) . Explain why the Mean Value Theorem does not apply over the interval [-1, 1]. 167. f(x)=|x12|For the following exercises, show there is no c such that f(1)f(1)=f(c)(2) . Explain why the Mean Value Theorem does not apply over the interval [-1, 1]. 168. f(x)=1x2 For the following exercises, show there is no c such that f(1)f(1)=f(c)(2) . Explain why the Mean Value Theorem does not apply over the interval [-1, 1]. 169. f(x)=|x|f(x)=[x](Hint: This is called the floor function and it is defined so that f(x) is the largest integer less than or equal to x.)For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 171. y=ex over [0, 1]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 172. y=In(2x+3)over[32,0]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 173. f(x)=tan(2x) over [0, 2]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 174. y=9x2 over [-3, 3]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 175. y=1|x+1| over [0, 3]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 176. y=x3+2x+1 over [0, 6]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 177. y=x2+3x+2x over [-1, 1]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 178. y=xsin(x)+1 over [0, 1]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 179. y=In(x+1) over [0, e - 1]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 180. y=xsin(x) over [0, 2]For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [a, b]. Justify your answer. 181. y=5+|x| over [-1, 1]For the following exercises, consider the roots of the equation. 82. Show that the equation y=x3+3x2+16 has exactly one real root. What is it?For the following exercises, consider the roots of the equation. 183. Find the conditions for exactly one root (double root) for the equation y=x2+bx+c .For the following exercises, consider the roots of the equation. 184. Find the conditions for y=exb have one root. Is it possible to have more than one root?For the following exercises, use a calculator to graph the function over the interval [a, b] and graph the secant line from a to b. Use the calculator to estimate all values of c as guaranteed by the Mean Value Theorem. Then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. 185. [T] y=tan(x)over[14,14]For the following exercises, use a calculator to graph the function over the interval [a, b] and graph the secant line from a to b. Use the calculator to estimate all values of c as guaranteed by the Mean Value Theorem. Then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. 186. [T] y=1x+1 over [0, 3]For the following exercises, use a calculator to graph the function over the interval [a, b] and graph the secant line from a to b. Use the calculator to estimate all values of c as guaranteed by the Mean Value Theorem. Then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. 187. [T] y=|x2+2x4| over [-4, 0]For the following exercises, use a calculator to graph the function over the interval [a, b] and graph the secant line from a to b. Use the calculator to estimate all values of c as guaranteed by the Mean Value Theorem. Then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. 188. [T] y=x+1xover[12,4]For the following exercises, use a calculator to graph the function over the interval [a, b] and graph the secant line from a to b. Use the calculator to estimate all values of c as guaranteed by the Mean Value Theorem. Then, find the exact value of c, if possible, or write the final equation and use a calculator to estimate to four digits. 189. [T] y=x+1+1x2 over [3, 8]At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.Show that y=sec2x and y=tan2x have the same derivative. What can you say about y=sec2xtan2x ?Show that y=csc2x and y=cot2x have the same derivative. What can you say about y=csc2xcot2x ?If c is a critical point of f(x), when is there no local maximum or minimum at c? Explain.For the function y=x3 , is x=0 both an inflection point and a local maximum/minimum?For the function y=x3 , is x=0 an inflection point?Is it possible for a point c to be both an inflection point and a local extrema of a twice differentiable function?Why do you need continuity for the first derivative test? Come up with an example.Explain whether a concave-down function has to cross y=0 for some value of x.Explain whether a polynomial of degree 2 can have an inflection point.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing or decreasing. 201.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing or decreasing. 202.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing or decreasing. 203.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing or decreasing. 204.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing or decreasing. 205.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing and decreasing and the minima and maxima are located. 206.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing and decreasing and the minima and maxima are located. 207.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing and decreasing and the minima and maxima are located. 208.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing and decreasing and the minima and maxima are located. 209.For the following exercises, analyze the graphs of f’, then list all intervals where f is increasing and decreasing and the minima and maxima are located. 210.For the following exercises, analyze the graphs of f’, then list all inflection points and intervals f that are concave up and concave down. 211.For the following exercises, analyze the graphs of f’, then list all inflection points and intervals f that are concave up and concave down. 212.For the following exercises, analyze the graphs of f’, then list all inflection points and intervals f that are concave up and concave down. 213.For the following exercises, analyze the graphs of f’, then list all inflection points and intervals f that are concave up and concave down. 214.For the following exercises, analyze the graphs of f’, then list all inflection points and intervals f that are concave up and concave down. 215.For the following exercises, draw a graph that satisfies the given specifications for the domain x=[3,3] . The function does not have to be continuous or differentiable. 216. f(x)0,f(x)0 over x1,3x0,f(x)=0 over 0x1 For the following exercises, draw a graph that satisfies the given specifications for the domain x=[3,3] . The function does not have to be continuous or differentiable. 217. f(x)0 over x2,3x1,f(x)0 over 1x2,f(x)0 for all x For the following exercises, draw a graph that satisfies the given specifications for the domain x=[3,3] . The function does not have to be continuous or differentiable. 218. f(x)0 over 1x1,f(x)0,3x1,1x3 , local maximum at x = 0, local maximum at x=2 For the following exercises, draw a graph that satisfies the given specifications for the domain x=[3,3] . The function does not have to be continuous or differentiable. 219. There is a local maximum at x = 2, local minimum at x = 1, and the graph is neither concave up nor concave down.For the following exercises, draw a graph that satisfies the given specifications for the domain x=[3,3] . The function does not have to be continuous or differentiable. 220. There are local maxima at x=1 , the function is concave up for all x, and the function remains positive for all x.For the following exercises, determine intervals where f is increasing or decreasing and local minima and maxima of f. 221. f(x)=sinx+sin3x over x For the following exercises, determine intervals where f is increasing or decreasing and local minima and maxima of f. 222. f(x)=x2+cosx For the following exercises. determine a. intervals where f is concave up or concave down, and b. the inflection points of f. 223. f(x)=x34x2+x+2

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