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All Textbook Solutions for Calculus (MindTap Course List)
71EA model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t)=60t, at which time the fuel is exhausted and it becomes a freely falling body. Fourteen seconds later, the rockets parachute opens, and the downward velocity slows linearly to 18 ft/s in 5 seconds. The rocket then floats to the ground at that rate. a Determine the position function s and the velocity function v for all times t. Sketch the graphs of 5 and v. b At what time does the rocket reach its maximum height, and what is that height? c At what time does the rocket land?73E1CC2CCa State Fermats Theorem. b Define a critical number of f.4CC5CC6CC7CC8CCIf you have a graphing calculator or computer, why do you need calculus to graph a function?10CC11CC1TFQ2TFQ3TFQ4TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or given an example that disproves the statement. If f(x)0 for 1x6, then f is decreasing on (1,6).6TFQ7TFQ8TFQ9TFQ10TFQ11TFQ12TFQ13TFQ14TFQ15TFQ16TFQ17TFQ18TFQ19TFQ1E2E1-6 Find the local and absolute extreme values of the function on the given interval. f(x)=3x4x2+1,[2,2]4E1-6 Find the local and absolute extreme values of the function on the given interval. f(x)=x+2cosx,[,]6E7E7-12 Find the limit. limtt3t+2(2t1)(t2+t+1)9E10E7-12 Find the limit. limx(4x2+3x2x)12E13-15 Sketch the graph of a function that satisfies the given conditions. f(0)=0,f(2)=f(1)=f(9)=0 limxf(x)=0,limxf(x)=, f(x)0on(,2),(1,6),and(9,), f(x)0on(2,1)and(6,9), f(x)0on(,0)and(12,), f(x)0on(0,6)and(6,12)14E15EThe figure shows the graph of the derivative f of a function f. a On what intervals is f increasing or decreasing? b For what values of x does f have a local maximum or minimum? c Sketch the graph of f. d Sketch a possible graph of f.17-28 Use the guidelines of Section 3.5 to sketch the curve. y=22xx318E17-28 Use the guidelines of Section 3.5 to sketch the curve. y=3x44x3+217-28 Use the guidelines of Section 3.5 to sketch the curve. y=x1x217-28 Use the guidelines of Section 3.5 to sketch the curve. y=1x(x3)222E23E24E25E26E27E28E29E30E31E32EShow that the equation 3x+2cosx+5=0 has exactly one real root.34EBy applying the Mean Value Theorem to the function f(x)=x1/5 on the interval 32, 33, show that 23352.0125For what values of the constants a and b is 1, 3 a point of inflection of the curve y=ax3+bx2?37EFind two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.Show that the shortest distance from the point (x1,y1) to the straight line Ax+By+C=0 is |Ax1+By1+C|A2+B240EFind the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.42E43E44E45E46EA hockey team plays in an arena with a seating capacity of 15, 000 spectators. With the ticket price set at 12, average attendance at a game has been 11, 000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?48E49E50E51E52E53E54E55-58 Find f. f(t)=2t3sint,f(0)=556E57E58E59E60E61E62E63EIn an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.A rectangular beam will be cut from a cylindrical log of radius 10 inches. a Show that the beam of maximal cross-sectional area is a square. b Four rectangular planks will be cut from the four sections of the log that remain after cutting the square beam. Determine the dimensions of the planks that will have maximal cross-sectional area. c Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.If a projectile is fired with an initial velocity v at an angle of inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola y=(tan)xg2v2cos2x202 a Suppose the projectile is fired from the base of a plane that is inclined at an angle , 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by R()=2v2cossin()gcos2 b Determine so that R is a maximum. c Suppose the plane is at an angle below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.67E68EShow that |sinxcosx|2 for all x2PShow that the inflection points of the curve y=(sinx)/x lie on the curve y2(x4+4)=4.Find the point on the parabola y=1x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.5PWater is flowing at a constant rate into a spherical tank. Let V(t) be the volume of water in the tank and H(t) be the height of the water in the tank at time t. a What are the meanings of V(t) and H(t) Are these derivatives positive, negative, or zero? b Is V"(t) positive, negative, or zero? Explain. c Let t1,t2 and t3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H"(t1),H"(t2), and H"(t3) positive, negative, or zero? Why?7P8P9PAn isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point (0,a) on the y-axis see the figure. Find the value of a that minimizes the lengths of the equal sides. You may be surprised that the result does not give an equilateral triangle..The line y=mx+b intersects the parabola y=x2 in points A and B. See the figure. Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB.12P13P14P15Pa Let ABC be a triangle with right angle A and hypotenuse a=|BC|.see the figure. If the inscribed circle touches the hypotenuse at D, show that |CD|=12(|BC|+|AC||AB|) b If =12C express the radius r of the inscribed circle in terms of a and . c If a is fixed and varies, find the maximum value of r.17PABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas that the triangle AEE can have.19PFor what values of c is there is a straight line that intersects the curve y=x4+cx3+12x25x+2 in four distinct points?One of the problems posed by the Marquis de lHospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C where dr, a rope of length l is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. See the figure. As 1Hospital argued, this happens when the distance |ED| is maximized. Show that when the system reaches equilibrium, the value of x is r4d(r+r2+8d2) Notice that this expression is independent of both W and l.22PAssume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely?A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. The figure shows the case n=4. Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1+n.1Ea Use six rectangles to find estimates of each type for the area under the given graph of f from x=0 to x=12. i L6 sample points are left endpoints ii R6, sample points are right endpoints iii M6 sample points are midpoints b Is L6 an underestimate or overestimate of the true area? c Is R6 an underestimate or overestimate of the true area? d Which of the numbers L6, R6, or M6 gives the best estimate? Explain.a Estimate the area under the graph of f(x)=1/x from x=1 to x=2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? b Repeat part a using left endpoints.4E5E6E7EEvaluate the upper and lower sums for f(x)=1+x2,1x1, with n = 3 and 4. Illustrate with diagrams like Figure 14.With a programmable calculator or a computer, it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. On a TI use the Is command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop. Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for n = 10, 30, 50, and 100. Then guess the value of the exact area. The region under y=x4 from 0 to 110ESome computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if xi* is a left or right endpoint. For instance, in Maple use leftbox, rightbox, left- sum, and rightsum. a If f(x)=1/(x2+1),0x1, find the left and right sums for n = 10, 30, and 50. b Illustrate by graphing the rectangles in part a. c Show that the exact area under f lies between 0.780 and 0.791.12EThe speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. ts 0 0.5 1.0 1.5 2.0 2.5 3.0 vft/s 0 6.2 10.8 14.9 18.1 19.4 20.2The table shows speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida. a Estimate the distance the race car traveled during this Time s Velocity mi/h 0 182.9 10 168.0 20 106.6 30 99.8 40 124.5 50 176.1 60 175.6 time period using the velocities at the beginning of the time intervals. b Give another estimate using the velocities at the end of the time periods. c Are your estimates in parts a and b upper and lower estimates? Explain.15E16EThe velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.18EIn someone infected with measles, the virus level N measured in number of infected cells per mL of blood plasma reaches a peak density at about t=12 days when a rash appears and then decreases fairly rapidly as a result of immune response. The area under the graph of Nt from t = 0 to t = 12 as shown in the figure is equal to the total amount of infection needed to develop symptoms measured in density of infected cells time. The function N has been modeled by the function f(t)=t(t21)(t+1) Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003. Date Deaths per day March 1 0.0079 March 15 0.0638 March 29 0.1944 April 12 0.4435 April 26 0.5620 May 10 0.4630 May 24 0.2897 a By using an argument similar to that in Example 4, estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. b How would you interpret the number of SARS deaths as an area under a curve? Source: A. Gumel et al., Modelling Strategies for Controlling SARS Outbreaks, Proceedings of the Royal Society of London: Series B 271 2004: 2223-32.Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=2xx2+1,1x3Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=x2+1+2x,4x7Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=sinx,0x24EDetermine a region whose area is equal to the given limit. Do not evaluate the limit. limni=1n4ntani4na Use Definition 2 to find an expression for the area under the curve y=x3 from 0 to 1 as a limit. b The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part a. 13+23+33+...+n3=[n(n1)2]227E28E29E30E31E32EEvaluate the Riemann sum for f(x)=x1,6x4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.If f(x)=cosx0x3/4 evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. Give your answer correct to six decimal places. What does the Riemann sum represent? Illustrate with a diagram.If f(x)=x24,0x3, find the Riemann sum with n = 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.a Find the Riemann sum for f(x)=1/x,1x2, with four terms, taking the sample points to be right endpoints. Give your answer correct to six decimal places. Explain what the Riemann sum represents with the aid of a sketch, b Repeat part a with midpoints as the sample points.The graph of a function f is given. Estimate 010f(x)dx using five subintervals with a right endpoints, b left endpoints, and c midpoints.The graph of g is shown. Estimate 24g(x)dx with six subintervals using a right endpoints, b left endpoints, and c midpoints.7EThe table gives the values of a function obtained from an experiment. Use them to estimate 39f(x)dx using three equal subintervals with a right endpoints, b left endpoints, and c midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral? x 3 4 5 6 7 8 9 f(x) 3.4 2.1 0.6 0.3 0.9 1.4 1.8Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 08sinxdx,n=4Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 01x3+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 02xx+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 2xsin2xdx,n=4If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles use RiemannSum or middlesum and middlebox commands in Maple, check the answer to Exercise 11 and illustrate with a graph. Then repeat with n = 10 and n = 20.With a programmable calculator or computer see the instructions for Exercise 4.1.9, compute the left and right Riemann sums for the function f(x)=x/(x+1) on the interval 0, 2with n = 100. Explain why these estimates show that 0.894602xx+1dx0.908115E16E17E18EExpress the limit as a definite integral on the given interval. limni=1n[5(x*i)34x*i]x,[2, 7]Express the limit as a definite integral on the given interval. limni=1nxi*(xi*)2+4x,[1,3]21E22EUse the form of the definition of the integral given in Theorem 4 to evaluate the integral. 20(x2+x)dx24E25E26EProve that abxdx=b2a22Prove that abx2dx=b3a3329EExpress the integral as a limit of Riemann sums. Do not evaluate the limit. 25(x2+1x)dx31EExpress the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. 210x6dxThe graph of f is shown. Evaluate each integral by interpreting it in terms of areas. a 02f(x)dx b 05f(x)dx c 57f(x)dx d 09f(x)dxThe graph of g consists of two straight fines and a semicircle. Use it to evaluate each integral. a 02g(x)dx b 26g(x)dx c 07g(x)dxEvaluate the integral by interpreting it in terms of areas. 12(1x)dxEvaluate the integral by interpreting it in terms of areas. 09(13x2)dx37E38EEvaluate the integral by interpreting it in terms of areas. 43|12x|dx40E41EGiven that 0sin4xdx=38, what is 0sin4d?43E44E45E46E47EIf 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.If 09f(x)dx=37 and 09g(x)dx=16, find 09[2f(x)+3g(x)]dxFind 05f(x)dx if f(x)={3forx3xforx3For the function / whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning. A 08f(x)dx B 03f(x)dx C 38f(x)dx D 48f(x)dx E f(1)If F(x)=2xf(t)dt, where f is the function whose graph is given, which of the following values is largest? A F0 B F1 C F2 D F3 E F4Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of 42[f(x)+2x+5]dxSuppose / has absolute minimum value m and absolute maximum value M. Between what two values must 02f(x)dx lie? Which property of integrals allows you to make your conclusion?Use the properties of integrals to verify the inequality without evaluating the integrals. 04(x24x+4)dx0Use the properties of integrals to verify the inequality without evaluating the integrals. 011+x2dx011+xdx57E58E59E60E61EUse Property 8 of integrals to estimate the value of the integral. 02(x33x+3)dx63E64EUse properties of integrals, together with Exercises 27 and 28, to prove the inequality. 13x4+1dx26366E67E68E69E70ELet f(x)=0 if x is any rational number and f(x)=1 if x is any irrational number. Show that f is not integrable on 0, 1.72EExpress the limit as a definite intergal. limni=1ni4n5Hint: Consider f(x)=x4.74EFind 12x2dx. Hint: Choose xi* to be the geometric mean of xi1 and xi that is, xi*=xi1xi and use the identity 1m(m+1)=1m1m+1Explain exactly what is meant by the statement that differentiation and integration are inverse processes.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. a Evaluate gx for x=0,1,2,3,4,5,and6 b Estimate g(7). c Where does g have a maximum value? Where does it have a minimum value? d Sketch a rough graph of g.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. a Evaluate g(0),g(1),g(2),g(3), and g(6). b On what interval is g increasing? c Where does g have a maximum value? d Sketch a rough graph of g.Let g(x)=0xf(t)dt where f is the function whose graph is shown. a Evaluate g(0) and g(6). b Estimate g(x) for x=1,2,3,4,and5. c On what interval is g increasing? d Where does g have a maximum value? e Sketch a rough graph of g. f Use the graph in part e to sketch the graph of g(x). Compare with the graph of f.Sketch the area represented by g(x). Then find g(x) in two ways: a by using Part 1 of the Fundamental Theorem and b by evaluating the integral using Part 2 and then differentiating. g(x)=1xt2dtSketch the area represented by g(x). Then find g(x) in two ways: a by using Part 1 of the Fundamental Theorem and b by evaluating the integral using Part 2 and then differentiating. g(x)=0x(2+sint)dt7EUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(x)=1xcos(t2)dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(s)=sx(tt2)8dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(u)=0utt+1dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x)=x01+sectdtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. R(y)=y2t3sintdtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=21/xsin4tdtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=1xz2z4+1dzUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=13x+2t1+t3dt16E17E18E19EEvaluate the integral. 11x100dx21E22E23E24EEvaluate the integral. /6sindEvaluate the integral. 55dx27EEvaluate the integral. 04(4t)tdtEvaluate the integral. 142+x2xdxEvaluate the integral. 12(3u2)(u+1)du31EEvaluate the integral. /4/3csc2d33EEvaluate the integral. 12s2+1s2dsEvaluate the integral. 12v5+3v6v4dv36E37E38E39E40E41E42E43E44E45E46EEvaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. 12x3dx48EWhat is wrong with the equation? 21x4dx=x33]21=38What is wrong with the equation? 124x3dx=2x2]12=3251EWhat is wrong with the equation? 0sec2xdx=tanx]0=0Find the derivative of the function. g(x)=2x3xu21u2+1du [Hint:2x3xf(u)du=2x0f(u)du+03xf(u)du]54E55E56ELet F(x)=xcosttdt. Find an equation of the tangent line to the curve y=F(x) at the point with x-coordinate .58EOn what interval is the curve y=0xt2t2+t+2dt concave downward?Let F(x)=1xf(t)dt, where f is the function whose graph is shown. Where is F concave downward?61E62EThe Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8. a At what values of x does this function have local maximum values? b On what intervals is the function concave upward? c Use a graph to solve the following equation correct to two decimal places: 0xsin(t2/2)dt=0.2The sine integral function Si(x)=0xsinttdt is important in electrical engineering. The integrand f(t)=(sint)/t is not defined when t=0, but we know that its limit is 1 when t0. So we define f(0)=1 and this makes f a continuous function everywhere. a Draw the graph of Si. b At what values of x does this function have local maximum values? c Find the coordinates of the first inflection point to the right of the origin. d Does this function have horizontal asymptotes? e Solve the following equation correct to one decimal place: 0xsinttdt=1Let g(x)=0xf(t)dt where f is the function whose graph is shown. a At what values of x do the local maximum and minimum values of g occur? b Where does g attain its absolute maximum value? c On what intervals is g concave downward? d Sketch the graph of g.Let g(x)=0xf(t)dt where f is the function whose graph is shown. a At what values of x do the local maximum and minimum values of g occur? b Where does g attain its absolute maximum value? c On what intervals is g concave downward? d Sketch the graph of g.Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on 0, 1. limni=1n(i4n5+in2)Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on 0, 1. limn1n(1n+2n+3n+...+nn)