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All Textbook Solutions for Calculus (MindTap Course List)
118E119ESignum Function The signum function is defined by sgn(x)={1,x00,x=01,x0 Sketch a graph of sgn(x) and find the following (if possible). (a) limx0sgn(x) (b) limx0+sgn(x) (c) limx0sgn(x)121ECreating Models A swimmer crosses a pool of width b by swimming in a straight line from (0, 0) to (2b, b). (See figure.) (a) Let f be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during the swimmers crossing of the pool. Determine the function f and sketch its graph. Is f continuous? Explain. (b) Let g be the minimum distance between the swimmer and the long sides of the pool. Determine the function g and sketch its graph. Is g continuous? Explain.Making a Function Continuous Find all values of c such that f is continuous on (,). f(x)={1x2,xcx,xc124E125E126E127E128E129E130EInfinite Limit In your own words, describe the meaning of an infinite limit. What does represent?2EDetermining Infinite Limits from a Graph In Exercises 3-6, determine whether f ( x ) approaches or as x approaches 2 from the left and from the right. f(x)=2|xx24|Determining Infinite Limits from a Graph In Exercises 3-6, determine whether f ( x ) approaches or as x approaches 2 from the left and from the right. f(x)=1x+2Determining Infinite Limits from a Graph In Exercises 3-6, determine whether f ( x ) approaches or as x approaches 2 from the left and from the right. f(x)=tanx46E7EDetermining Infinite Limits from a Graph In Exercises 7-10, determine whether f ( x ) approaches or as r approaches 4from the left and from the right. f(x)=1x4Determining Infinite Limits from a Graph In Exercises 7-10, determine whether f ( x ) approaches or as r approaches 4from the left and from the right. f(x)=1(x4)2Determining Infinite Limits from a Graph In Exercises 7-10, determine whether f ( x ) approaches or as r approaches 4from the left and from the right. f(x)=1(x4)2Numerical and Graphical Analysis In Exercises 11-16, create a table of values for the function and use the result to determine whether f ( x ) approaches to or as r approaches 3 from the left and from the right. Use a graphing utility to graph the function to confirm your answer. f(x)=1x29Numerical and Graphical Analysis In Exercises 11-16, create a table of values for the function and use the result to determine whether f(x) approaches to or as x approaches 3 from the left and from the right. Use a graphing utility to graph the function to confirm your answer. f(x)=xx2913E14E15E16EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. f(x)=1x218EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. f(x)=x2x2420EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. g(t)=t1t2+122EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. f(x)=3x2+x224E25E26E27EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. h(t)=t22tt416Finding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. f(x)=cscx30EFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. s(t)=tsintFinding Vertical Asymptotes In Exercises 17-32. find the vertical asymptotes (if any) of the graph of the function. g()=tan33E34E35E36EFinding a One-Sided Limit In Exercises 37-50, find the one-sided limit (if it exists). limx2+xx238EFinding a One-Sided Limit In Exercises 37-50, find the one-sided limit (if it exists). limx3x+3x2+x640E41E42EFinding a One-Sided Limit In Exercises 37-50, find the one-sided limit (if it exists). limx4(x2+2x+4)Finding a One-Sided Limit In Exercises 37-50, find the one-sided limit (if it exists). limx0+(x1x+3)45EFinding a One-Sided Limit In Exercises 37-50, find the one-sided limit (if it exists). limx(/2)+2cosx47E48E49E50E51E52E53E54E55E56E57ERelativity According to the theory of relativity, the mass m of a particle depends on its velocity v. That is, m=m01(v2/c2), where m0 is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass as v approaches c from the left.59E60ERate of Change A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, then the top will move down the wall at a rate of r=2x625x2ft/sec where x is the distance between the base of the ladder and the house, and r is the rate in feet per second. (a) Find the rate r when x is 7 feet. (b) Find the rate r when x is 15 feet. (c) Find the limit of r as x approaches 25 from the left.Average Speed On a trip of d miles to another city, a truck drivers average speed was x miles per hour. On the return trip, the average speed was y miles per hour. The average speed for the round trip was 50 miles per hour. (a) Verify that y=25xx25. What is the domain? (b) Complete the table. x 30 40 50 60 y Are the values of y different than you expected? Explain. (c) Find the limit of y as x approaches 25 from the right and interpret its meaning.Numerical and Graphical Analysis Consider the shaded region outside the sector of a circle of radius 10 meter: and inside a right triangle (see figure). (a) Write the area A= f() of the region as a function of . Determine the domain of the function. (b) Use a graphing utility to complete the table and graph the function over the appropriate domain. 0.3 0.6 0.9 1.2 1.5 f() (c) Find the limit of A as approaches /2 from the left.Numerical and Graphical Reasoning A crossed belt connects a 20centimeter pulley (10cm radius) on an electric motor with a 40centimeter pulley (20cm radius) on a saw arbor (see figure). The electric motor runs at 1700 revolutions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt. Write L as a function of . where is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) Use a graphing utility to complete the table. 0.3 0.6 0.9 1.2 1.5 L (e) Use a graphing utility to graph the function over the appropriate domain. (f) Find lim(/2) L. (g) Use a geometric argument as the basis of a second method of finding the limit in part (f). (h) Find lim0 L.65ETrue or False? In Exercises 65-68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of polynomial functions have no vertical asymptotes.True or False? In Exercises 65-68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of trigonometric functions have no vertical asymptotes.68EFinding Functions Find functions f and g such that limx0f(x)= and limx0g(x)=, but limx0[f(x)g(x)]070E71E72E73E74E75E76EFinding the Derivative by the Limit Process In Exercises 1-4, find the derivative of the function by the limit process. f(x)=12Finding the Derivative by the Limit Process In Exercises 1-4, Find the derivative of the function by the limit process. f(x)=5x4Finding the Derivative by the Limit Process In Exercises 1-4, find the derivative of the function by the limit process. f(x)=x32x+14RE5RE6REDetermining Differentiability In Exercises 7 and 8, describe the x-values at which f is differentiable. f(x)=(x3)2/58RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24REVibrating String When a guitar string is plucked, it vibrates with a frequency of F=200T. where F is measured in vibrations per second and the tension T is measured in pounds. Find live rates of change of the frequency when (a) T=4 pounds and (b) T=9 pounds.26REVertical Motion In Exercises 27 and 28, use the position function s(t)=16t2+v0t+s0 for free-falling objects. A ball is thrown straight down from the top of a 600-foot building with an initial velocity of 30 feet per second. (a) Determine the position and velocity functions for the ball. (b) Determine the average velocity on the interval [1, 3]. (c) Find the instantaneous velocities when t=1 and t=3. (d) Find the time required for the ball to reach ground level. (e) Find the velocity of the ball at impact.28RE29REFinding a Derivative In Exercises 29-40, use the Product Rule or the Quotient Rule to find the derivative of the function. g(x)=(2x3+5x)(3x4)31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48REFinding a Second Derivative In Exercises 4552, find the second derivative of the function. f()=3tan50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77REHarmonic Motion The displacement from equilibrium of an object in harmonic motion on the end of a spring is y=14cos8t14sin8t where y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when t=/479RE80RE81RE82RE83RE84RE85RE86RE87RE88RE89RE90RE1PSFinding Equations of Tangent Lines Graph the two parabolas y=x2andy=x2+2x5 in the same coordinate plane. Find equations of the two lines that are simultaneously tangent to both parabolas.Finding a Polynomial Find a third-degree polynomial p(x) that is tangent to the line y=14x13 at the point (1,1), and tangent to the line y=2x5 at the point (1,3).4PSTangent Lines and Normal Lines (a) Find an equation of the tangent line to the parabola y=x2 at the point (2,4). (b) Find an equation of the normal line to y=x2 at the point (2,4). (The normal line at a point is perpendicular to the tangent line at the point.) Where does this line intersect the parabola a second time? (c) Find equations of the tangent line and normal line to y=x2 at the point (0,0). (d) Prove that for any point (a, b) (0,0) on the parabola y=x2, the normal line intersects the graph a second time.Finding Polynomials (a) Find the polynomial P1(x)=a0+a1x whose value and slope agree with the value and slope of f(x)=cosx at the point x=0. (b) Find the polynomial P2(x)=a0+a1x+a2x2 whose value and first two derivatives agree with the value and first two derivatives of f(x)=cosx at the point x=0. This polynomial is called the second-degree Taylor polynomial of f(x)=cosxatx=0. (c) Complete the table comparing the values of f(x)=cosxandP2(x). What do you observe? x -1.0 -0.1 -0.001 0 0.001 0.1 1.0 cosx P2(x) (d) Find the third-degree Taylor polynomial of f(x)=sinxatx=0.7PS8PSShadow Length A man 6 feet tall walks at a rate of 5 feet per second toward a streetlight that is 30 feet high (see figure). The mans 3-foot-tall child follows at the same speed, but 10 feet behind the man. The shadow behind the child is caused by the man at some times and by the child at other times. (a) Suppose the man is 90 feet from the streetlight. Show that the mans shadow extends beyond the childs shadow. (b) Suppose the man is 60 feet from the streetlight. Show that the childs shadow extends beyond the mans shadow. (c) Determine the distance d from the man to the streetlight at which the tips of the two shadows are exactly the same distance from the streetlight. (d) Determine how fast the tip of the mans shadow is moving as a function of x, the distance between the man and the streetlight. Discuss the continuity of this shadow speed function.Moving Point A particle is moving along the graph of y=x3 (see figure). When x=8, the y-component of the position of the particle is increasing at the rate of 1 centimeter per second. (a) How fast is the x-component changing at this moment? (b) How fast is the distance from the origin changing at this moment? (c) How fast is the angle of inclination changing at this moment?11PSProof Let E be a function satisfying E(0)=E(0)=1. Prove that if E(a+b)=E(a)E(b) for all a and b, then E is differentiable and E(x)=E(x) for all x. Find an example of a function satisfying E(a+b)=E(a)E(b)13PS14PSAcceleration and Jerk If a is the acceleration of an object, then the jerk j is defined by j=a(t). (a) Use this definition to give a physical interpretation of j. (b) Find j for the slowing vehicle in Exercise 119 in Section 2.3 and interpret the result. (c) The figure shows the graphs of the position, velocity, acceleration, and jerk functions of a vehicle. Identify each graph and explain your reasoning.Tangent Line Describe how to find the slope of the tangent line to the graph of a function at a point.Notation List four notation alternatives to f(x).3E4EEstimating Slope In Exercises 5 and 6, estimate the slope of the graph at the points (x1,y1) and (x2,y2)Estimating Slope In Exercises 5 and 6, estimate the slope of the graph at the points (x1,y1) and (x2,y2)Slopes of Secant Lines In Exercises 7 and 8, use the graph shown in the figure. To print an enlarged copy of file graph, go to MalhGraphs.com. Identify or sketch each of the quantities on the figure. (a) f(1)andf(4) (b) f(4)f(1) (c) 41 (d) y2=f(4)f(1)41(x1)Slopes of Secant Lines In Exercises 7 and 8. use the graph shown in the figure. To print an enlarged copy of file graph, go to MathGraphs.com. Insert the proper inequality symbol ( or ) between the given quantities. (a) f(4)f(1)41f(4)f(3)43 (b) f(4)f(1)41f(1)Finding the Slope of a Tangent Line In Exercises 9-14, find the slope of the tangent line to the graph of the function at the given point. f(x)=35x,(1,8)Finding the Slope of a Tangent Line In Exercises 9-14, find the slope of the tangent line to the graph of the function at the given point. g(x)=32x+1,(2,2)11EFinding the Slope of a Tangent Line In Exercises 9-14, find the slope of the tangent line to the graph of the function at the given point. f(x)=5x2,(3,4)Finding the Slope of a Tangent Line In Exercises 9-14, find the slope of the tangent line to the graph of the function at the given point. f(t)=3tt2,(0,0)Finding the Slope of a Tangent Line In Exercises 9-14, find the slope of the tangent line to the graph of the function at the given point. h(t)=t2+4t,(1,5)Finding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. f(x)=7Finding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. g(x)=317EFinding the Derivative by the Limit Process In Exercises 15-28, find the derivative of the function by the limit process. f(x)=7x3Finding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. h(s)=3+23s20EFinding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. f(x)=x2+x3Finding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. f(x)=x2523E24EFinding the Derivative by the Limit Process In Exercises 15-28, Find the derivative of the function by the limit process. f(x)=1x126E27E28EFinding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=x2+3,(1,4)30EFinding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=x3,(2,8)32E33EFinding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=x1,(5,2)Finding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=x+4x,(4,5)Finding an Equation of a Tangent Line In Exercises 29-36, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=x1x,(1,0)37EFinding an Equation of a Tangent Line In Exercises 37-42. find an equation of the line that is tangent to the graph of f and parallel to the given line. Function Line f(x)=2x2 4x+y+3=0Finding an Equation of a Tangent Line In Exercises 37-42. find an equation of the line that is tangent to the graph of f and parallel to the given line. Function Line f(x)=x3 3xy+1=0Finding an Equation of a Tangent Line In Exercises 37-42. find an equation of the line that is tangent to the graph of f and parallel to the given line. Function Line f(x)=x3+2 3xy4=041E42E43E44E45E46ESketching a Derivative In Exercises 43-48, sketch the graph of fExplain how you found your answer.48ESketching a Graph Sketch a graph of a function whose derivative is always negative. Explain how you found the answer.Sketching a Graph Sketch a graph of a function whose derivative is zero at exactly two points. Explain how you found the answer.51E52E53EUsing a Tangent Line The tangent line to the graph of y=h(x) at the point (-1, 4) passes through the point (3, 6).Find h(1) and h(1).Working Backwards In Exercises 55-58, the Unlit represents f(c) Tor a function f and a number c . Find f and c. lim0[53(1+x)]2x56E57EWorking Backwards In Exercises 55-58, the Unlit represents f(c) Tor a function f and a number c . Find f and c. limx92x6x959E60E61EFinding an Equation of a Tangent Line In Exercises 61 and 62, find equations of the two tangent lines to the graph off that pass through the indicated point. f(x)=x263EHOW DO YOU SEE IT? The figure shows die graph of g. (a) g(0)= (b) g(3)= (c) What can you conclude about the graph of g knowing that g(1)=83? (d) What can you conclude about the graph of g knowing that g(4)=73? (e) Is g(6)g(4) positive or negative? Explain. (f) Is it possible to find g(2) from die graph? Explain.65E66E67E68EUsing the Alternative Form of the Derivative In Exercises 69-76, use the alternative form of the derivative to find the derivative at x=c, if it exists. f(x)=x3+2x2+1,c=270E71E72E73E74E75EUsing the Alternative Form of the Derivative In Exercises 69-76, use the alternative form of the derivative to find the derivative at x=c, if it exists. f(x)=x6,c=6Determining Differentiability In Exercises 77-80, describe the x -values at which f is differentiable. f(x)=(x+4)2/3Determining Differentiability In Exercises 77-80, describe the x -values at which f is differentiable. f(x)=x2x2479EDetermining Differentiability In Exercises 77-80, describe the x -values at which f is differentiable. f(x)={x24,x04x2,x081E82E83E84E85EDetermining Differentiability In Exercises 85-88, find the derivatives from the left and from the right at x=1 (if they exist). Is the function differentiable at x=1? f(x)=1x287E88E89EDetermining Differentiability In Exercises 89 and 90, determine whether the function is differentiable at x=2. f(x)={12x+2,x22x,x291EConjecture Consider the functions f(x)=x2 and g(x)=x3. (a) Graph f and f on the same set of axes. (b) Graph g and g on the same set of axes. (c) Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h (x) if h(x)=xn. where " is an integer and n2 ? (d) Find f'(x) if f(x)=x4 Compare the result will) the conjecture in part (c). Is this a proof of your conjecture? Explain.True or False? In Exercises 93-96, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the tangent line to the differentiable function f at the point (2, f (2)) is f(2+x)f(2)x.94E95ETrue or False? In Exercises 93-96. determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.Differentiability and Continuity Let f(x)={xsin1x,x00,x=0 and g(x)={x2sin1x,x00,x=0. Show that f is continuous, but not differentiable, at x=0.Show that e is differentiable at g and find g(0).98E1E2E3E4EEstimating Slope In Exercises 5 and 6, use the graph to estimate the slope of the tangent line to y=xn at the point (1.1). Verify your answer analytically. To print an enlarged copy of the graph, go to Math Graph.com. (a) y=x1/2 (b) y=x3Estimating Slope In Exercises 5 and 6, use the graph to estimate the slope of the tangent line to y=xn at the point (1.1). Verify your answer analytically. To print an enlarged copy of the graph, go to Math Graph. com. (a) y=x1/2 (b) y=x1Finding a Derivative In Exercises 7-26. Use the rules of differentiation to find the derivative of the function. y=12Finding a Derivative In Exercises 7-26. Use the rules of differentiation to find the derivative of the function. f(x)=9