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All Textbook Solutions for Calculus: Early Transcendentals
(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 2012.74EWhen a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v of the air-stream is related to the radius r of the trachea by the equation v(r)=k(r0r)r212r0rr0 where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 12r0 is prevented (otherwise the person would suffocate). (a) Determine the value of r in the interval [12r0,r0] at which v has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of v on the interval? (c) Sketch the graph of v on the interval [0, r0].Show that 5 is a critical number of the function g(x)=2+(x5)3 but g does not have a local extreme value at 5.Prove 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.A 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.(a) Suppose that f is differentiable on and has two roots. Show that f has at least one root. (b) Suppose f is twice differentiable on and has three roots. Show that f has at least one real root. (c) Can you generalize parts (a) and (b)?If f(1) = 10 and f(x) 2 for 1 x 4, how small can f(4) possibly be?Suppose that 3 f(x) 5 for all values of x. Show that 18 f(8) f(2) 30.Does there exist a function f such that f(0) = 1, f(2) = 4, and f(x) 2 for all x?Suppose 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 a x b. Prove that f(b) g(b). [Hint: Apply the Mean Value Theorem to the function h = f g.]Show that sin x x if 0 x 2.Suppose 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.Use the Mean Value Theorem to prove the inequality sina=sinbabforallaandbIf f(x) = c (c a constant) for all x, use Corollary 7 to show that f(x) = cx + d for some constant d.Let f(x) = l/x and g(x)={1xifx01+1xifx0 Show that f'(x) = g(x) for all x in their domains. Can we conclude from Corollary 7 that f g is constant?Use the method of Example 6 to prove the identity 2sin1x=cos1(12x2)x0At 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.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.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?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?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 + 4(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) = 2x3 9x2 + 12x 3(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) = x4 2x2 + 3(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+1(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) = sin x + cos x, 0 x 2(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) = cos2x 2 sin x, 0 x 2(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) = e2x + ex(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) = x2 ln x(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) = x2 x ln x(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) = x4exFind the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x) = 1 + 3x2 2x3Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x)=x2x1Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x)=xx4(a) Find the critical numbers of f(x) = x4(x 1)3. (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you?Suppose f is continuous on (, ). (a) If f(2) = 0 and f(2) = 5, what can you say about f? (b) If f(6) = 0 and f(6) = 0, what can you say about f?Sketch the graph of a function that satisfies all of the given conditions. (a) f(x) 0 and f(x) 0 for all x (b) f(x) 0 and f(x) 0 for all xSketch the graph of a function that satisfies all of the given conditions. (a) f(x) 0 and f(x) 0 for all x (b) f(x) 0 and f(x) 0 for all xSketch the graph of a function that satisfies all of the given conditions. Vertical asymptote x = 0, f(x) 0 if x 2, f(x) 0 if x 2 (x 0), f(x) 0 if x 0, f(x) 0 if x 0Sketch the graph of a function that satisfies all of the given conditions. f(0) = f(2) = f(4) = 0, f(x) 0 if x 0 or 2 x 4, f(x) 0 if 0 x 2 or x 4, f(x) 0 if 1 x 3, f(x) 0 if x 1 or x 3Sketch the graph of a function that satisfies all of the given conditions. f(x) 0 for all x 1, vertical asymptote x = 1, f(x) 0 if x 1 or x 3, f(x) 0 if 1 x 3Sketch the graph of a function that satisfies all of the given conditions. f(5) = 0, f(x) 0 when x 5, f(x) 0 when x 5, f(2) = 0, f(8) = 0, f(x) 0 when x 2 or x 8, f(x) 0 for 2 x 8, limxf(x)=3, limxf(x)=3Sketch the graph of a function that satisfies all of the given conditions. f(0) =f(4) = 0, f(x) = 1 if x 1, f(x) 0 if 0 x 2, f(x) 0 if 1 x 0 or 2 x 4 or x 4, limx2f(x)=, limx2+f(x)= f(x) 0 if 1 x 2 or 2 x 4, f(x) 0 if x 4Sketch 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 248E(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+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)=x2+1x(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)=ex1ex(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)=ex2(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)=x16x223lnx55E56ESuppose the derivative of a function f is f(x) = (x + 1)2 (x 3)5 (x 6)4. On what interval is f increasing?58E59E60E61E62EIn 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.67ELet f(t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) f(3) = 2, f(3) = 4 (b) f(3) = 2, f(3) = 4 (c) f(3) = 2, f(3) = 4 (d) f(3) = 2, f(3) = 469E70E71E72EFind 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.For what values of the numbers a and b does the function f(x)=axebx2 have the maximum value f(2) = 1?75E76E77E78E79E80EAssume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and g is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesnt the argument in parts (a) and (b) work in this case?82E83E(a) Show that ex 1 + x for x 0. (b) Deduce that ex1+x+12x2forx0. (c) Use mathematical induction to prove that for x 0 and any positive integer n, ex1+x+x22!++xnn!Show 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)/3.86E87E88EShow that the function g(x) = x | x | has an inflection point at (0, 0) but g(0) does not exist.90EFor what values of c is the function f(x)=cx+1x2+3 increasing on (, )?The 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 x 0, f(x)=x4sin1xg(x)=x(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.Given 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)Given 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) limxa[f(x)p(x)] (b) limxa[h(x)p(x)] (c) limxa[p(x)q(x)]Given 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) limxa[f(x)p(x)] (b) limxa[p(x)q(x)] (c) limxa[p(x)+q(x)]Given 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) limxa[f(x)]g(x) (b) limxa[f(x)]p(x) (c) limxa[h(x)]p(x) (d) limxa[p(x)]f(x) (e) limxa[p(x)]q(x) (f) limxap(x)q(x)Use 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?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. limx3x3x29Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx4x22x8x4Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx2x3+8x+2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1x32x2+1x31Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1/26x2+5x44x2+16x9Find 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(/2)+cosx1sinxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0tan3xsin2x15EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0x21cosx17EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. lim1+cos1cosFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxlnxxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxx+x212x2Find 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+lnxxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxlnxx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limt1t81t5124EFind 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/10u3Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0ex1xx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0sinhxxx3Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0tanhxtanxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0xsinxxtanxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0sin1xxFind 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(lnx)2xFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0x3x3x1Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0cosmxcosnxx2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0ln(1+x)cosx+ex1Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx1xsin(x1)2x2x1Find 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+arctan(2x)lnxFind 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+xx1lnx+x139EFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0exex2xxsinxFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0cosx1+12x2x4Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxa+cosxln(xa)ln(exea)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. limxxsin(/x)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. limxxex/2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limx0sin5xcsc3xFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxxln(11x)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. limxx3ex2Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxx3/2sin(1/x)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+lnxtan(x/2)50EFind 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(xx11lnx)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(cscxcotx)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+(1x1ex1)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+(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)xFind 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(12x)1/xFind 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(1+ax)bxFind 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+x1/(1x)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. limxx(ln2)/(1+lnx)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. limxx1/xFind the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why. limxxexFind 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+(4x+1)cotx66EFind 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+(1+sin3x)1/x68EUse a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. limx(1+2x)xUse a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. limx05x4x3x2xIllustrate lHospitals Rule by graphing both f(x)/g(x) and f(x)/g(x) near x = 0 to see that these ratios have the same limit as x 0. Also, calculate the exact value of the limit. f(x) = ex 1, g(x) = x3 + 4xIllustrate lHospitals Rule by graphing both f(x)/g(x) and f(x)/g(x) near x = 0 to see that these ratios have the same limit as x 0. Also, calculate the exact value of the limit. f(x) = 2x sin x, g(x) = sec x 1Prove that limxexxn= for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x.Prove that limxlnxxp=0 for any number p 0. This shows that the logarithmic function approaches infinity more slowly than any power of x.What happens if you try to use lHospitals Rule to find the limit? Evaluate the limit using another method. limxxx2+1What happens if you try to use lHospitals Rule to find the limit? Evaluate the limit using another method. limx(/2)secxtanxInvestigate the family of curves f(x) = ex cx. In particular, find the limits as x and determine the values of c for which f has an absolute minimum. What happens to the minimum points as c increases?If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is v=mgc(1ect/m) where g is the acceleration due to gravity and c is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; c is the proportionality constant.) (a) Calculate limtv. What is the meaning of this limit? (b) For fixed t, use lHospitals Rule to calculate limc0+v. What can you conclude about the velocity of a falling object in a vacuum?79ELight enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (See the figure.) A light beam A that enters through the center of the pupil measures brighter than a beam B entering near the edge of the pupil. They detailed their findings of this phenomenon, known as the StilesCrawford effect of the first kind, in an important paper published in 1933. In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil as they expected. The percentage P of the total luminance entering a pupil of radius r mm that is sensed at the retina can be described by P=110r2r2ln10 where is an experimentally determined constant, typically about 0.05. (a) What is the percentage of luminance sensed by a pupil of radius 3 mm? Use = 0.05. (b) Compute the percentage of luminance sensed by a pupil of radius 2 mm. Does it make sense that it is larger than the answer to part (a)? (c) Compute limx0+P. Is the result what you would expect? Is this result physically possible? Source: Adapted from W. Stiles and B. Crawford, The Luminous Efficiency of Rays Entering the Eye Pupil at Different Points. Proceedings of the Royal Society of London, Series B: Biological Sciences 112 (1933): 42850.Some populations initally grow exponentially but eventually level off. Equations of the form P(t)=M1+Aekt Where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and A=MP0P0, where P0 is the initial population. (a) Compute limtP(t). Explain why your answer is to be expected. (b) Compute limMP(t). (Note that A is defined in terms of M.) What kind of function is your result?82E83EThe figure shows a sector of a circle with central angle . Let A() be the area of the segment between the chord PR and the arc PR. Let B() be the area of the triangle PQR. Find lim0+A()/B().Evaluate limx[xx2ln(1+xx)].Suppose f is a positive function. If limxaf(x)=0 and limxag(x)=, show that limxa[f(x)]g(x)=0 This shows that 0 is not an indeterminate form.If f is continuous, f(2) = 0, and f(2) = 7, evaluate limx0f(2+3x)+f(2+5x)xFor what values of a and b is the following equation true? limx0(sin2xx3+a+bx2)=0If f is continuous, use lHospitals Rule to show that limh0f(x+h)f(xh)2h=f(x) Explain the meaning of this equation with the aid of a diagram.90ELet f(x)={e1/x2ifx00ifx=0 (a) Use the definition of derivative to compute f(0). (b) Show that f has derivatives of all orders that are defined on . [Hint: First show by induction that there is a polynomial pn(x) and a nonnegative integer kn such that f(n)(x)=pn(x)f(x)/xkn for x 0.)Let f(x)={xxifx01ifx=0 (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f. (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?Use the guidelines of this section to sketch the curve. y = x3 + 3x2Use the guidelines of this section to sketch the curve. y = 2 + 3x2 x3Use the guidelines of this section to sketch the curve. y = x4 4xUse the guidelines of this section to sketch the curve. y = x4 8x2 + 8Use the guidelines of this section to sketch the curve. y = x(x 4)3Use the guidelines of this section to sketch the curve. y = x5 5xUse the guidelines of this section to sketch the curve. y=15x583x3+16xUse the guidelines of this section to sketch the curve. y = (4 x2)5Use the guidelines of this section to sketch the curve. y=xx1Use the guidelines of this section to sketch the curve. y=x2+5x25x2Use the guidelines of this section to sketch the curve. y=xx223x+x2Use the guidelines of this section to sketch the curve. y=1+1x+1x2Use the guidelines of this section to sketch the curve. y=xx24Use the guidelines of this section to sketch the curve. y=1x24Use the guidelines of this section to sketch the curve. y=x2x2+3Use the guidelines of this section to sketch the curve. y=(x1)2x2+1Use the guidelines of this section to sketch the curve. y=x1x2Use the guidelines of this section to sketch the curve. y=xx31Use the guidelines of this section to sketch the curve. y=x3x3+1Use the guidelines of this section to sketch the curve. y=x3x2Use the guidelines of this section to sketch the curve. y=(x3)xUse the guidelines of this section to sketch the curve. y=(x4)x3Use the guidelines of this section to sketch the curve. y=x2+x2Use the guidelines of this section to sketch the curve. y=x2+xxUse the guidelines of this section to sketch the curve. y=xx2+1Use the guidelines of this section to sketch the curve. y=x2x2The table gives the population of the world P(t), in millions, where t is measured in years and t = 0 corresponds to the year 1900. t Population (millions) 0 1650 10 1750 20 1860 30 2070 40 2300 50 2560 60 3040 70 3710 80 4450 90 5280 100 6080 110 6870 (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (c) Use your model in part (b) to find a model for the rate of population growth. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) In Section 1.1 we modeled P(t) with the exponential function f(t)=(1.43653109)(1.01395)t Use this model to find a model for the rate of population growth. (f) Use your model in part (e) to estimate the rate of growth in 1920 and 1980. Compare with your estimates in parts (a) and (d). (g) Estimate the rate of growth in 1985.Use the guidelines of this section to sketch the curve. y=xx2129EUse the guidelines of this section to sketch the curve. y = x5/3 5x2/3Use the guidelines of this section to sketch the curve. y=x21332EUse the guidelines of this section to sketch the curve. y = sin3xUse the guidelines of this section to sketch the curve. y = x + cos xUse the guidelines of this section to sketch the curve. y = x tan x, /2 x /236EUse the guidelines of this section to sketch the curve. y=sinx+3cosx, 2 x 238E39EUse the guidelines of this section to sketch the curve. y=sinx2+cosxUse the guidelines of this section to sketch the curve. y = arctan(ex)42EUse the guidelines of this section to sketch the curve. y = 1/(1 + ex)Use the guidelines of this section to sketch the curve. y = ex sin x, 0 x 2Use the guidelines of this section to sketch the curve. y=1x+lnx46E47E48E49EUse the guidelines of this section to sketch the curve. y = ln(1 + x3)Use the guidelines of this section to sketch the curve. y = xe1/x52E53EUse the guidelines of this section to sketch the curve. y=tan1(x1x+1)55E56EA model for the spread of a rumor is given by the equation p(t)=11+aekt where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants. (a) When will half the population have heard the rumor? (b) When is the rate of spread of the rumor greatest? (c) Sketch the graph of p.A model for the concentration at time t of a drug injected into the bloodstream is C(t)=K(eatebt) where a, b, and K are positive constants and b a. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes?The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve y=W24EIx4+WL12EIx3WL224EIx2 where E and I are positive constants. (E is Youngs modulus of elasticity and I is the moment of inertia of a cross-section of the beam.) Sketch the graph of the deflection curve.Coulombs Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge 1 at a position x between them. It follows from Coulombs Law that the net force acting on the middle particle is F(x)=kx2+k(x2)20x2 where k is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?Find an equation of the slant asymptote. Do not sketch the curve. y=x2+1x+1Find an equation of the slant asymptote. Do not sketch the curve. y=4x310x211x+1x23xFind an equation of the slant asymptote. Do not sketch the curve. y=2x35x2+3xx2x264EUse the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=x2x1Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=1+5x2x2x267EUse the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=x3(x+1)2Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y=1+12x+ex