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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I
37E38E39E40E41E42E43E44EA formula for the derivative of a function f is given. How many critical numbers does f have? f(x) = 5e0.1|x| sin x 1A formula for the derivative of a function f is given. How many critical numbers does f have? f(x)=100cos2x10+x21Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 12 + 4x x2, [0, 5]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 5 + 54x 2x3, [0, 4]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 2x3 3x2 12x + 1, [2, 3]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x3 6x2 + 5, [3, 5]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 3x4 4x3 12x2 + 1, [2, 3]Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = (t2 4)3, [2, 3]Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x+1x,[0.2,4]Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=xx2x+1,[0,3]Find the absolute maximum and absolute minimum values of f on the given interval. f(t)=tt3,[1,4]Find the absolute maximum and absolute minimum values of f on the given interval. f(t)=t1+t2,[0,2]Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2cos t + sin 2t, [0, /2]Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = t + cot (t/2), [/4, 7/4]Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x2lnx,[12,4]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = xex/2, [3, 1]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = ln(x2 + x + 1), [1, 1]Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = x 2 tan1 x, [0, 4]If a and b are positive numbers, find the maximum value of f(x) = xa(1 x)b, 0 x 1.Use a graph to estimate the critical numbers of f(x) = |1 + 5x x3| correct to one decimal place.65E(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. f(x) = ex + e2x, 0 x 1(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. f(x)=xxx2(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. f(x) = x 2 cos x, 2 x 0After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function C(t)=1.35te2802t models the average BAC, measured in mg/mL, of a group of eight male subjects t hours after rapid consumption of 15 mL of ethanol (corresponding to one alcoholic drink). What is the maximum average BAC during the first 3 hours? When does it occur?After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t)=8(e0.4te0.6t) where the time t is measured in hours and C is measured in g/mL. What is the maximum concentration of the antibiotic during the first 12 hours?Between 0C and 30C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula V=999.870.06426T+0.0085043T20.0000679T3 Find the temperature at which water has its maximum density.An 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)=0.01441t30.4177t2+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 415175E76E77E78E79EA cubic function is a polynomial of degree 3; that is, it has the form f(x) = ax3 + bx2 + cx + d, where a 0. (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?The graph of a function f is shown. Verify that f satisfies the hypotheses of Rolles Theorem on the interval [0, 8]. Then estimate the value(s) 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 value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [0, 8]. (c) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6].Draw the graph of a function that is continuous on [0, 8] where f(0) = 1 and f(8) = 4 and that does not satisfy the conclusion of the Mean Value Theorem on [0, 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) = 2x2 4x + 5, [1, 3]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) = x3 2x2 4x + 2, [2, 2]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]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) = x + 1/x, [12,2]Let f(x) = 1 x2/3. Show that f(l) = f(1) but there is no number c in (1, 1) such that f(c) = 0. Why does this not contradict Rolles Theorem?Let f(x) = tan x. Show that f(0) = f() but there is no number c in (0, ) such that f(c) = 0. Why does this not contradict Rolles Theorem?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) = 2x2 3x + 1, [0, 2]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 3x + 2, [2, 2]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) = ln x, [1, 4]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) = 1/x, [1, 3]Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function. the secant line through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel? f(x)=x, [0, 4]Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function. the secant line through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel? f(x) = ex, [0, 2]Let f(x) = (x 3)2. Show that there is no value of c in (1, 4) such that f(4) f(1) = f(c)(4 1). Why does this not contradict the Mean Value Theorem?Let f(x) = 2 |2x 1|. Show that there is no value of c such that f(3) f(0) = f(c)(3 0). Why does this not contradict the Mean Value Theorem?Show that the equation has exactly one real root. 2x + cos x = 0Show that the equation has exactly one real root. x3 + ex = 0Show that the equation x3 15x + 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.24E25E26EDoes there exist a function f such that f(0) = 1, f(2) = 4, and f(x) 2 for all x?28EShow that sin x x if 0 x 2.30E31EIf f(x) = c (c a constant) for all x, use Corollary 7 to show that f(x) = cx + d for some constant d.33EUse the method of Example 6 to prove the identity 2sin1x=cos1(12x2)x036E37EA 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.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.2E3E4EThe 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?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?(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) = x3 3x2 9x + 410E11E(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)=xx2+113E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30ESketch the graph of a function that satisfies all of the given conditions. f(x) 0 if x 2, f(x) 0 if x 2, f(x) 0 if x 2, f has inflection point (2, 5), limxf(x)=8, limxf(x)=032ESuppose f is a continuous function where f(x) 0 for all x, f(0) = 4, f(x) 0 if x 0 or x 2, f(x) 0 if 0 x 2, f(l) = f(l) = 0, f(x) 0 if x 1 or x 1, f(x) 0 if 1 x 1. (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 point(s) are the following true? (a) dydxandd2ydx2arebothpositive. (b) dydxandd2ydx2arebothnegative. (c) dydxisnegativebutd2ydx2ispositive.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-coordinate(s) of the point(s) of inflection. (e) Assuming that f(0) = 0, sketch a graph of f.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-coordinate(s) of the point(s) of inflection. (e) Assuming that f(0) = 0, sketch a graph of f.(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) = x3 12x + 2(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) = 36x + 3x2 2x3(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)=12x44x2+3(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 + x4(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)5 5x 2(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) = 5x3 3x5(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)=x6x(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) = 5x2/3 2x5/3(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. C(x) = x1/3(x + 4)(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) = ln(x2 + 9)(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() = 2 cos + cos2, 0 2(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) = x sin x, 0 x 4(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x)=1+1x1x2(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x)=x24x2+451E(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of f. f(x)=ex1ex53E54E55E56E57E58E59E60E61E62EIn 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.67E68E69E70E71E72EFind a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at x = 2 and a local minimum value of 0 at x = 1.74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E92E93EGiven that limxaf(x)=0limxag(x)=0limxah(x)=1limxap(x)=limxaq(x)= which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. (a) limxaf(x)g(x) (b) limxaf(x)p(x) (c) limxah(x)p(x) (d) limxap(x)f(x) (e) limxap(x)q(x)2E3E4EUse the graphs of f and g and their tangent lines at (2, 0) to find limx2f(x)g(x).Use the graphs of f and g and their tangent lines at (2, 0) to find limx2f(x)g(x).The graph of a function f and its tangent line at 0 are shown. What is the value of limx0f(x)ex1?8E9E10E11E12E13E14E15E16E17E18E19E20EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+lnxx22E23E24EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx01+2x14xxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limueu/10u327E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0(cscxcotx)53EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+(1x1tan1x)Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx(xlnx)Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1+[ln(x71)ln(x51)]Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+xxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0+(tan2x)x59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78EIf an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A=A0(1+rn)nt If we let n , we refer to the continuous compounding of interest. Use 1Hospitals Rule to show that if interest is compounded continuously, then the amount after t years is A=A0ert80E81E82E83E84E85E86E87E88E89E90E91E92EUse the guidelines of this section to sketch the curve. y = x3 + 3x22E3E4E5E6E7E8E9EUse the guidelines of this section to sketch the curve. y=x2+5x25x211EUse the guidelines of this section to sketch the curve. y=1+1x+1x213E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E