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All Textbook Solutions for Calculus (MindTap Course List)
58E59EIf f and g are the functions whose graphs are shown, let P(x)=f(x)g(x), P(x)=f(x)g(x), and C(x)=f(g(x)).Find a P(2),b Q(2), and c C(2)61E62E63E64E65E66E67E68E69E6971 Find h in terms of f.and g. h(x)=f(x)g(x)71E72EA particle moves on a vertical line so that its coordinate at time r is y=t312t+3,t0. a Find the velocity and acceleration functions. b When is the particle moving upward and when is it moving downward? c Find the distance that the particle travels in the time interval 0t3. d Graph the position, velocity, and acceleration functions for 0t3. e When is the particle speeding up? When is it slowing down?The volume of a right circular cone is V=13r2h, where r is the radius of the base and h is the height. a Find the rate of change of the volume with respect to the height if the radius is constant. b Find the rate of change of the volume with respect to the radius if the height is constant.The mass of part of a wire is x(1+x) kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x=4m.The cost, in dollars, of producing x units of a certain commodity is C(x)=920+2x0.02x2+0.00007x3 a Find the marginal cost function. b Find C'100 and explain its meaning. c Compare C'100 with the cost of producing the 101st item.77E78E79EA waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp?81E82Ea Find the linearization of f(x)=1+3x3 at a=0. State the corresponding linear approximation and use it to give an approximate value for 1.033. b Determine the values of x for which the linear approximation given in part a is accurate to within 0.1.84EA window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.8688 Express the limit as a derivative and evaluate. limx1x171x187E88E89E90E91EShow that the length of the portion of any tangent line to the asteroid x2/3+y2/3=a2/3 cut of by the coordinate axes is constant.Find points P and Q on the parabola y=1x2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle. See the figure.2PShow that the tangent lines to the parabola y=ax2+bx+c at any two points with x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p and q.4P5P6P7P8P9P10PHow many lines are tangent to both of the circles x2+y2=4 and x2+(y3)2=1? At what points do these tangent lines touch the circles?If f(x)=x46+x45+21+x, calculate f(46)(3). Express your answer using factorial notation:n!=1.2.3.....(n1).n.The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. a Find the angular velocity of the connecting rod, d/dt, in radius per sound, when =/3. b Express the distance x=|OP| in terms of . c Find an expression for the velocity of the pin P in terms of .Tangent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y=x2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2; it intersects T1 at Q1 and T2 at Q2. Show that |PQ1||PP1|+|PQ2||PP2|=1Let T and N be the tangent and normal lines to the ellipse x2/9+y2/4=1 at any point P on the ellipse in the first quadrant. Let xT and yT be the x- and y-intercepts of T and xN and yN be the intercepts of N. As P moves along the ellipse in the first quadrant but not on the axes, what values can xT,yT,xN, and yN take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is.16P17PLet P(x1,y1) be a point on the parabola y2=4px with focus F(p,0). Let e the angle between the parabola and the line segment FP, and let be the angle between the horizontal line y=y1 and the parabola as in the figure. Prove that =. Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.Suppose that we replace the parabolic mirror of problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ will be reflected to the point R on the axis so that PQO=OQR the angle of incidence is equal to the angle of reflection. What happens to the point R as P is taken closer and closer to the axis?20P21PGiven an ellipse x2/a2+y2/b2=1, where ab, find the equation of the set of all points from which there are two tangents to the curve whose slopes are a reciprocals and b negative reciprocals.23P24PA lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 25 intersects some of these circles.26P27P28PExplain the difference between an absolute minimum and a local minimum.Suppose f is a continuous function defined on a closed interval [a,b] a What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f? b What steps would you take to find those maximum and minimum values?For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.Use the graph to state the absolute and local maximum and minimum values of the function.Use the graph to state the absolute and local maximum and minimum values of the function.Sketch the graph of a function f that is continuous on 1, 5 and has the given properties. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4Sketch the graph of a function f that is continuous on 1, 5 and has the given properties. Absolute maximum at 4, absolute minimum at 5, local maximum at 2, local minimum at 39ESketch the graph of a function f that is continuous on 1, 5 and has the given properties. Absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or minimum there.11E12Ea Sketch the graph of a function on [1,2] that has an absolute maximum but no absolute minimum. b Sketch the graph of a function on [1,2] that is discontinuous but has both an absolute maximum and an absolute minimum.14ESketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. Use the graphs and transformations of Sections 1.2 and 1.3. f(x)=12(3x1),x316E17E18ESketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. Use the graphs and transformations of Sections 1.2 and 1.3. f(x)=sinx,0x/220E21E22ESketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. Use the graphs and transformations of Sections 1.2 and 1.3. f(x)=1+(x+1)2,2x524ESketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. Use the graphs and transformations of Sections 1.2 and 1.3. f(x)=1x26E27E28E29E30E31E32E33E34EFind the critical numbers of the function. g(y)=y1y2y+136EFind the critical numbers of the function. h(t)=t3/42t1/438EFind the critical numbers of the function. F(x)=x4/5(x4)240EFind the critical numbers of the function. f()=2cos+sin242E43E44EFind the absolute maximum and absolute minimum values of f on the given interval. f(x)=12+4xx2,[0,5]46E47E48E49E50EFind the absolute maximum and absolute minimum values of f on the given interval. f(x)=x+1x,[0.2,4]52EFind the absolute maximum and absolute minimum values of f on the given interval. f(t)=tt3,[1,4]54EFind the absolute maximum and absolute minimum values of f on the given interval. f(t)=2cost+sin2t,[0,/2]56EIf a and b are positive numbers, find the maximum value of f(x)=xa(1x)b,0x1.58E59E60E61E62E63EAn object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is F=Wsin+cos where is a positive constant called the coefficient of friction and where 0/2. Show that F is minimized when tan=.The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function L(t)=0.0144t30.4177t2+2.703t+1060.1 where t is measured in months since January 1, 2012. Estimate when the water level was highest during 2012.On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. a Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval t 0, 125. Then graph this polynomial. b Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds. Event Time s Velocity ft/s Launch 0 0 Begin roll maneuver 10 185 End roll maneuver 15 319 Throttle to 89 20 447 Throttle to 67 32 742 Throttle to 104 59 1325 Maximum dynamic pressure 62 1445 Solid rocket booster separation 125 415167E68EProve that the function f(x)=x101+x51+x+1 has neither a local maximum nor a local minimum.If f has a local minimum value at c, show that the function g(x)=f(x) has a local maximum value at c.Prove Fermats Theorem for the case in which f has a local minimum at c.72EThe graph of a function f is shown. Verify that f satisfies the hypotheses of Rolles Theorem on the interval 0, 8. Then estimate the values of c that satisfy the conclusion of Rolles Theorem on that interval.Draw the graph of a function defined on 0, 8 such that f(0)=f(8)=3 and the function does not satisfy the conclusion of Rolles Theorem on 0, 8.The graph of a function g is shown. a Verify that g satisfies the hypotheses of the Mean Value Theorem on the interval 0, 8. b Estimate the values of c that satisfy the conclusion of the Mean Value Theorem on the interval 0, 8. c Estimate the values of c that satisfy the conclusion of the Mean Value Theorem on the interval 2, 6.4E5E6E5-8 Verify that the function satisfies the three hypotheses of Rolles Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolles Theorem. f(x)=sin(x/2),[/2,3/2]8ELet f(x)=1x2/3. Show that f(1)=f(1) but there is no number c in (1,1) such that fc = 0. Why does this not contradict Rolles Theorem?Let f(x)=tanx. Show that f(0)=f() but there is no number c in (0,) such that fc = 0. Why does this not contradict Rolles Theorem?11-14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f(x)=2x23x+1,[0,2]12E11-14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f(x)=x3,[0,1]14E15E15-16 Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant fine through the endpoints, and the tangent line at c, fc. Are the secant line and the tangent line parallel? f(x)=x32x,[2,2]Let f(x)=(x3)2. Show that there is no value of c in 1, 4 such that f(4)f(1)=f(c)(41). Why does this not contradict the Mean Value Theorem?Let f(x)=2|2x1|. Show that there is no value of c such that f(3)f(0)=f(c)(30). Why does this not contradict the Mean Value Theorem?19-20 Show that the equation has exactly one real root. 2x+cosx=020EShow that the equation x315x+c=0 has at most one root in the interval [2,2].Show that the equation x4+4x+c=0 has at most two real roots.a Show that a polynomial of degree 3 has at most three real roots. b Show that a polynomial of degree n has at most n real roots.24EIf f(1)=10 and f(x)2 for 1x4, how small can f(4) possibly be?26E27ESuppose that f and g are continuous on a, b and differentiable on a, b. Suppose also that f(a)=g(a) and f(x)g(x) for axb. Prove that f(b)g(b). Hint: Apply the Mean Value Theorem to the function h=fg.29ESuppose f is an odd function and is differentiable everywhere. Prove that for every positive number b, there exists a number c in (b,b) such that f(c)=f(b)/b.31E32E33EAt 2:00 pm a cars speedometer reads 30 mi/h. At 2:10 pm it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. Hint: Consider f(t)=g(t)h(t), where g and h are the position functions of the two runners.A number a is called a fixed point of a function f if f(a)=a. Prove that if f(x)1 for all real numbers x, then f has at most one fixed point.1-2 Use the given graph of f to find the following. a The open intervals on which f is increasing. b The open intervals on which f is decreasing. c The open intervals on which f is concave upward. d The open intervals on which f is concave downward. e The coordinates of the points of inflection.1-2 Use the given graph of f to find the following. a The open intervals on which f is increasing. b The open intervals on which f is decreasing. c The open intervals on which f is concave upward. d The open intervals on which f is concave downward. e The coordinates of the points of inflection.Suppose you are given a formula for a function f. a How do you determine where f is increasing or decreasing? b How do you determine where the graph of f is concave upward or concave downward? c How do you locate inflection points?a State the First Derivative Test. b State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?5-6 The graph of the derivative f of a function f is shown. a On what intervals is f increasing or decreasing? b At what values of xdoes f have a local maximum or minimum?5-6 The graph of the derivative f of a function f is shown. a On what intervals is f increasing or decreasing? b At what values of x does f have a local maximum or minimum?In each part state the x-coordinates of the inflection points of f. Give reasons for your answers. a The curve is the graph of f. b The curve is the graph of f'. c The curve is the graph of f''.The graph of the first derivative f of a function f is shown. a On what intervals is f increasing? Explain. b At what values of x does f have a local maximum or minimum? Explain. c On what intervals is f concave upward or concave downward? Explain. d What are the x-coordinates of the inflection points of f? Why?9E10E11E12E9-14 a Find the intervals on which f is increasing or decreasing. b Find the local maximum and minimum values of f. c Find the intervals of concavity and the inflection points. f(x)=sinx+cosx,0x214E15-17 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x)=1+3x22x316E15-17 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x)=xx418E19E20E21E22E23E24E25E20-27 Sketch the graph of a function that satisfies all of the given conditions. f(0)=f(4)=0,f(x)=1ifx1,f(x)0if0x2,f(x)0if1x0or2x4orx4,limx2f(x)=,limx2+f(x)=f(x)0if1x2or2x4,f(x)0ifx427E28ESuppose f is a continuous function where f(x)0 for all x, f(0)=4,f(x)0ifx0orx2,f(x)0if0x2,f(1)=f(1)=0,f(x)0ifx1orx1,f(x)0if1x1. a Can f have an absolute maximum? If so, sketch a possible graph of f, If not, explain why. b Can f have an absolute minimum? If so, sketch a possible graph of f. If not, explain why. c Sketch a possible graph for f that does not achieve an absolute minimum.The graph of a function y=f(x) is shown. At which points are the following true? a dydx and d2ydx are both positive. b dydx and d2ydx are both negative. c dydx is negative but d2ydx is positive.31-32 The graph of the derivative f of a continuous function f is shown. a On what intervals is f increasing? Decreasing? b At what values of x does f have a local maximum? Local minimum? c On what intervals is f concave upward? Concave downward? d State the x-coordinates of the points of inflection. e Assuming that f(0)=0, sketch a graph of f.31-32 The graph of the derivative f of a continuous function f is shown. a On what intervals is f increasing? Decreasing? b At what values of x does f have a local maximum? Local minimum? c On what intervals is f concave upward? Concave downward? d State the x-coordinates of the points of inflection. e Assuming that f(0)=0, sketch a graph of f.33E34E35E33-44 a Find the intervals of increase or decrease. b Find the local maximum and minimum values. c Find the intervals of concavity and the inflection points. d Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. g(x)=200+8x3+x433-44 a Find the intervals of increase or decrease. b Find the local maximum and minimum values. c Find the intervals of concavity and the inflection points. d Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. h(x)=(x+1)55x238E33-44 a Find the intervals of increase or decrease. b Find the local maximum and minimum values. c Find the intervals of concavity and the inflection points. d Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. F(x)=x6x40E41E42E33-44 a Find the intervals of increase or decrease. b Find the local maximum and minimum values. c Find the intervals of concavity and the inflection points. d Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. f()=2cos+cos2,0233-44 a Find the intervals of increase or decrease. b Find the local maximum and minimum values. c Find the intervals of concavity and the inflection points. d Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one. S(x)=xsinx,0x445E46E47E47-48 a Use a graph of f to estimate the maximum and minimum values. Then find the exact values. b Estimate the value of x at which f increases most rapidly. Then find the exact value. f(x)=x+2cosx,0x249E50E51E52EA graph of a population of yeast cells in a new laboratory culture as a function of time is shown. a Describe how the rate of population increase varies. b When is this rate highest? c On what intervals is the population function concave upward or downward? d Estimate the coordinates of the inflection point.In an episode of The Simpsons television show, Homer reads from a newspaper and announces Heres good news According to this eye-catching article, SAT scores are declining at a slower rate. Interpret Homers statement in terms of a function and its first and second derivatives.55E56E57E58E59E60Ea If the function f(x)=x3+ax2+bx has the local minimum value 293 at x=13, what are the values of a and b? b Which of the tangent fines to the curve in part a has the smallest slope?62E63E64E65E66E67E68EShow that a cubic function a third-degree polynomial always has exactly one point of inflection. If its graph has three x-intercepts x1,x2, and x3, show that the x-coordinate of the inflection point is (x1+x2+x3)/3.For what values of c does the polynomial P(x)=x4+cx3+x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?Prove that if (c,f(c)) is a point of inflection of the graph of f and f exists in an open interval that contains c, then f(c)=0. Hint: Apply the First Derivative Test and Fermats Theorem to the function g=f.Show that if f(x)=x4, then f(0)=0, but 0, 0 is not an inflection point of the graph of f.Show that the function g(x)=x|x| has an inflection point at 0, 0 but g(0) does not exist.74E75E76EThe three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for x0, f(x)=x4sin1xg(x)=x4(2+sin1x)h(x)=x4(2+sin1x) a Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. b Show that f has neither a local maximum nor a local minimum at 0, g has a local minimum, and h has a local maximum.Explain in your own words the meaning of each of the following. a limxf(x)=5 b limxf(x)=3a Can the graph of y=f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. b How many horizontal asymptotes can the graph of y=f(x) have? Sketch graphs to illustrate the possibilities.For the function f whose graph is given, state the following. a limxf(x) b limxf(x) c limx1f(x) d limx3f(x) e The equations of the asymptotesFor the function g whose graph is given, state the following. a limxg(x) b limxg(x) c limx0g(x) d limx2g(x) e limx2+g(x) f The equations of the asymptotes5E6E7E7-8 Evaluate the limit and justify each step by indicating the appropriate properties of limits. limx9x3+8x435x+x39E10E11E12E9-32 Find the limit or show that it does not exist. limxt+t22tt29-32 Find the limit or show that it does not exist. limxttt2t3/2+3t59-32 Find the limit or show that it does not exist. limx(2x2+1)2(x1)2(x2+x)9-32 Find the limit or show that it does not exist. limxx2x4+19-32 Find the limit or show that it does not exist. limx1+4x62x39-32 Find the limit or show that it does not exist. limx1+4x62x39-32 Find the limit or show that it does not exist. limxx+3x24x19-32 Find the limit or show that it does not exist. limxx+3x24x19-32 Find the limit or show that it does not exist. limx(9x2+x3x)9-32 Find the limit or show that it does not exist. limx(4x2+3x+2x)9-32 Find the limit or show that it does not exist. limx(x2+axx2+bx)9-32 Find the limit or show that it does not exist. limxcosx9-32 Find the limit or show that it does not exist. limxx43x2+xx3x+29-32 Find the limit or show that it does not exist. limxx2+19-32 Find the limit or show that it does not exist. limx(x2+2x7)9-32 Find the limit or show that it does not exist. limx1+x6x4+19-32 Find the limit or show that it does not exist. limx(xx)9-32 Find the limit or show that it does not exist. limx(x2x4)9-32 Find the limit or show that it does not exist. limxxsin1x9-32 Find the limit or show that it does not exist. limxxsin1xa Estimate the value of limx(x2+x+1+x) by graphing the function f(x)=x2+x+1+x. b Use a table of values of f(x) to guess the value of the limit. c Prove that your guess is correct.a Use a graph of f(x)=3x2+8x+63x2+3x+1 to estimate the value of limxf(x) to one decimal place. b Use a table of values of fx to estimate the limit to four decimal places. c Find the exact value of the limit.35-40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y=5+4xx+335-40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y=2x2+13x2+2x135-40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y=2x2+x1x2+x235-40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y=1+x4x2x435-40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. y=x3xx26x+540E41E42ELet P and Q be polynomials. Find limxP(x)Q(x) if the degree of P is a less than the degree of Q and b greater than the degree of Q.44EFind a formula for a function / that satisfies the following conditions: limxf(x)=0,limx0f(x)=,f(2)=0, limx3f(x)=,limx3+f(x)=46EA function f is a ratio of quadratic functions and has a vertical asymptote x=4 and just one x-intercept, x=1.It is known that f has a removable discontinuity at x=1 and limx1f(x)=2 Evaluate a f(0) b limxf(x)48E48-51 Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. y=1x1+x48-51 Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. y=xx2+151E52E