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All Textbook Solutions for Calculus (MindTap Course List)
Fixed Point In Exercises 25 and 26, approximate the fixed point of the function to two decimal places. [A fixed point of a function f is a real number c such that f(c)=c.] f(x)=cotx,0,x27E28E29E30E31E32EMechanics Rule The Mechanics Rule for approximating a,a0, is xn+1=12(xn+axn),n=1,2,3, where x, is an approximation of a. (a) Use Newtons Method and the function f(x)=x2a to derive the Mechanics Rule. (b) Use the Mechanics Rule to approximate 5 and 7 to three decimal places.Approximating Radicals (a) Use Newtons Method and the function f(x)=xna to obtain a general rule for approximating xan. (b) Use the general rule found in part (a) to approximate 64 and 153 to three decimal places.Approximating Reciprocals Use Newtons Method to show that the equation xn+1=xn(2axn) can be used to approximate 1/a when x1 is an initial guess of the reciprocal of a. Note that this method of approximating reciprocals uses only the operations of multiplication and subtraction. (Hint: Consider f(x)=1xa).36E37ETrue or False? In Exercises 3740, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the coefficients of a polynomial function are all positive, then the polynomial has no positive zeros.True or False? In Exercises 3740, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f (x) is a cubic polynomial such that f(x) is never zero, then any initial guess will force Newtons Method to coverage to the zero of f.True or False? In Exercises 3740, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Newtons Method fails when the initial guess x1 corresponds to a horizontal tangent line for the graph of f at x1.Tangent Lines The graph of f(x)=sinx has infinitely many tangent lines that pass through the origin. Use Newtons Method to approximate to three decimal places the slope of the tangent line having the greatest slope.Point of Tangency The graph of f(x)=cosx and a tangent line to f through (he origin are shown. Find the coordinates of the point of tangency to three decimal places.CONCEPT CHECK Tangent Line Approximations What is the equation of the tangent line approximation to the graph of a function f at the point (c, f(c))?2E3ECONCEPT CHECK Finding Differentials Explain how to find a differential of a function.5E6E7E8E9EUsing a Tangent Line Approximation In Exercises 510, find the tangent line approximation T to the graph of f at the given point. Then complete the table. x 1.9 1.99 2 2.01 2.1 f(x) T(x) f(x)=cscx,(2,csc2)Verifying a Tangent Line Approximation In Exercises 11 and 12, verify the tangent line approximation of the function at the given point. Then use a graphing utility to graph the function and its approximation in the same viewing window. FunctionApproximationPoint f(x)=x+4 y=2+x4 (0, 2)12E13E14E15E16E17E18EFinding a Differential In Exercises 1928, find the differential dy of the given function. y=3x2420E21E22E23E24E25E26E27E28EUsing Differentials In Exercises 29 and 30, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to MathGraphs.com.Using Differentials In Exercises 29 and 30, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to MathGraphs.com.31EUsing Differentials In Exercises 31 and 32, use differentials and the graph of g' to approximate (a) g(2.93) and (b) g(3.1) given that g(3)=8.Area The measurement of the side of a square floor tile is 10 inches, with a possible error of 132 inch. (a) Use differentials to approximate the possible propagated error in computing the area of the square. (b) Approximate the percent error in computing the area of the square.Area The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. (a) Use differentials to approximate the possible propagated error in computing the area of the triangle. (b) Approximate the percent error in computing the area of the triangle.Volume and Surface Area The measurement of the edge of a cube is found to be 15 inches, with a possible error of 0.03 inch. (a) Use differentials to approximate the possible propagated error in computing the volume of the cube. (b) Use differentials to approximate the possible propagated error in computing the surface area of the cube. (c) Approximate the percent errors in parts (a) and (b).Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch. (a) Use differentials to approximate the possible propagated error in computing the volume of the sphere. (b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere. (c) Approximate the percent errors in parts (a) and (b).Stopping Distance The total stopping distance T of a vehicle is T = 2.5x + 0.5x2 where T is in feet and x is the speed in miles per hour. Approximate the change and percent change in total stopping distance as speed changes from x = 25 to x = 26 miles per hour.38EPendulum The period of a pendulum is given by T=2Lg where L is the length of the pendulum in feet, g is the acceleration due to gravity, and T is the time in seconds. The pendulum has been subjected to an increase in temperature such that the length has increased by 12. (a) Find the approximate percent change in the period by. (b) Using the result in part (a), find the approximate error in this pendulum clock in 1 day.40EProjectile Motion The range R of a projectile is R=v0232(sin2) where v0 is the initial velocity in feet per second and is the angle of elevation. Use differentials to approximate the change in the range when v0=2500 feet per second and is changed from 10 to 11.Surveying A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 71.5�. How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%?Approximating Function Values In Exercises 4346, use differentials to approximate the value of the expression. Compare your answer with that of a calculator. 99.444E45E46E47EUsing Defferentials Give a short explanation of why each approximation is valid. (a) 4.022+14(0.02) (b) tan0.050+1(0.05)49E50E51E52E53EFinding an Indefinite Integral In Exercises 1-8, find the indefinite integral. (x3+4)dxFinding an Indefinite Integral In Exercises 1-8, find the indefinite integral. (x4+3)dx3RE4RE5RE8RE6RE7RE9RE10RE11RE12REVertical Motion A ball is thrown vertically upward from ground level with an initial velocity of 96 feet per second. Assume the acceleration of the ball is a(t)=32 feet per second per second. (Neglect air resistance.) (a) How long will it take the ball to rise to its maximum height? What is the maximum height? (b) After how many seconds is the velocity of the ball one-half the initial velocity? (c) What is the height of the ball when its velocity is one-half the initial velocity?Vertical Motion With what initial velocity must an object be thrown upward (from a height of 3 meters) to reach a maximum height of 150 meters? Assume the acceleration of the object is a(t)=9.8 meters per second per second. (Neglect air resistance.)15RE16RE17RE18RE21RE22RE23RE24RE19RE20RE27REFinding Upper and Lower Sums for a Region In Exercises 27 and 28, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Function Interval f(x)=7x2 [0,3]29RE30RE31RE32RE25RE26RE33RE34RE35RE36RE37RE38RE39RE40RE43RE46RE41RE42RE49RE52RE44RE45RE53RE54RE55RE56RE47RE48RE57REUsing the Second Fundamental Theorem of Calculus In Exercises 57 and 58, use the Second Fundamental Theorem of Calculus to find F'(x). F(x)=1x1t2dt50RE51REFinding an Indefinite Integral In Exercises 59-66, find the indefinite integral. 6x33x4+2dx59RE63RE61RE62RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE1PSParabolic Arch Archimedes showed that the area of a parabolic arch is equal to 23 the product of the base and the height (see figure). (a) Graph the parabolic arch bounded by y = 9 x2 and the x-axis. Use an appropriate integral to find the area A. (b) Find the base and height of the arch and verify Archimedes formula. (c) Prove Archimedes formula for a general parabola.14PS5PSApproximation TheTwo-Point Gaussian Quadrature Approximation for f is 11f(x)f(13)+f(13). (a) Use this formula to approximate 11cosxdx. Find the error of the approximation. (b) Use this formula to approximate 1111+x2dx. (c) Prove that the Two-Point Gaussian Quadrature Approximation is exact for all polynomials of degree 3 or less.Extrema and Points of Inflection The graph of the function f consists of the three line segments joining the points (0, 0), (2,2), (6, 2), and (8, 3). The function F is defined by the integral F(x)=0xf(t)dt (a) Sketch the graph of f. (b) Complete the table. x 0 1 2 3 4 5 6 7 8 F(x) (c) Find the extrema of F on the interval [0, 8]. (d) Determine all points of inflection of F on the interval (0, 8).8PS9PS10PS11PS12PS13PSVelocity and Acceleration A car travels in a straight line for 1 hour. Its velocity v in miles per hour at six-minute intervals is shown in the table. t (hours) 0 0.1 0.2 0.3 0.4 0.5 v (mi/h) 0 10 20 40 60 50 t (hours) 0.6 0.7 0.8 0.9 1.0 v (mi/h) 40 35 40 50 65 (a) Produce a reasonable graph of the velocity function v by graphing these points and connecting them with a smooth curve. (b) Find the open intervals over which the acceleration a is positive. (c) Find the average acceleration of the car (in miles per hour per hour) over the interval [0, 0.4]. (d) What does the integral 01v(t)dx. signify? Approximate this integral using the Midpoint Rule with five subintervals. (e) Approximate the acceleration at t = 0.816PS17PS3PS4PSSine Integral Function The sine integral function Si(x)=0xsinttdt is often used in engineering. The function f(t)=sintt is not defined at t = 0, but its limit is 1 as t? 0. So, define f(0) = 1. Then f is continuous everywhere. (a) Use a graphing utility to graph Si(x). (b) At what values of x does Si(x) have relative maxima? (c) Find the coordinates of the first inflection point where x 0. (d) Decide whether Si(x) has any horizontal asymptotes. If so, identify each.19PS20PS21PSCONCEPT CHECK Antiderivative What does it mean for a function F to be an antiderivative of a function f on an interval I?Antiderivatives Can two different functions both be antiderivatives of the same function? Explain.Particular Solution What is a particular solution of a differential equation?4EIntegration and Differentiation In Exercises 5 and 6, verify the statement by showing that the derivative of the right side equals the integrand on the left side. (6x4)dx=2x3+CIntegration and Differentiation In Exercises 5 and 6, verify the statement by showing that the derivative of the right side equals the integrand on the left side. (8x3+12x2)dx=2x412x+CSolving a Differential Equation In Exercises 7-10, find the general solution of the differential equation and cheek the result by differentiation. dydt=9t28E9ESolving a Differential Equation In Exercises 7-10, find the general solution of the differential equation and cheek the result by differentiation. dydx=2x311ERewriting Before Integrating In Exercises 11-14, complete the table to find the indefinite integral. Original Integral Rewrite Integrate Simplify 14x2dxRewriting Before Integrating In Exercises 11-14, complete the table to Find the indefinite integral. Original Integral Rewrite Integrate Simplify 1xxdx14E15EFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (x5+1)dxFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (13x)dx19E20EFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (9x82x6)dxFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (x+1)(3x2)dx28E23E24E25EFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. x43x2+5x4dxFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (5cosx+4sinx)dx30E21E22E33E34EFinding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differentiation. (tan2y+1)dy36E31E32EFinding a Particular Solution In Exercises 37-44, find the particular solution of the differential equation that satisfies the initial condition(s). f(x)=6x,f(0)=8Finding a Particular Solution In Exercises 37-44, find the particular solution of the differential equation that satisfies the initial condition(s). g(x)=4x2,g(1)=339E40E41EFinding a Particular Solution In Exercises 37-44, find the particular solution of the differential equation that satisfies the initial condition(s). f(x)=3x2,f(1)=2,f(2)=343E44ESlope Field In Exercises 45 and 46, a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation, (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the indicated point. (To print an enlarged copy of the graph, go to MathGraphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point. dy/dx=x21,(1,3)46E47E48EEXPLORING CONCEPTS Sketching a Graph In Exercises 49 and 50, the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to MalhGraphs.com.Sketching a Graph In Exercises 49 and 50, the graph of the derivative of a function is given. Sketch the graphs of two functions that have the given derivative. (There is more than one correct answer.) To print an enlarged copy of the graph, go to MathGraphs.com.51EHOW DO YOU SEE IT? Use the graph of f shown in the figure to answer the following. (a) Approximate the slope of f at x=4. Explain. (b) Is f(5)f(4)0? Explain. (c) Approximate the value of x where f is maximum Explain. (d) Approximate any open intervals on which the graph of f is concave upward and any open intervals on which it is concave downward. Approximate the coordinates of any points of inflection.Horizontal Tangent Find a function f such that the graph of f has a horizontal tangent at (2, 0) and f(x)=2x.54ETree Growth An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dh/dt=1.5t+5, where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t=0). (a) Find the height after t years. (b) How tall are the shrubs when they are sold?Population Growth The rate of growth dP/dt of a population of bacteria is proportional to the square root of t, where P is the population size and t is the time in days (0t10). That is, dPdt=kt The initial size of the population is 500. After I day, the population has grown to 600. Estimate the population after 7 days.Vertical Motion In Exercises 57-59, assume the acceleration of the object is a(t)=32 feet per second per second. (Neglect air resistance.) A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 60 feet per second. How high will the ball go?Vertical Motion In Exercises 57-59, assume the acceleration of the object is a(t)=32 feet per second per second. (Neglect air resistance.) With what initial velocity must an object be thrown upward (from ground level) to reach the top of the Washington Monument (approximately 550 feet)?59EVertical Motion In Exercises 60-62, assume the acceleration of the object is a(t)=9.8 meters per second per second. (Neglect air resistance.) A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.61E62ELunar Gravity On the moon, the acceleration of a free-falling object is a(t)=1.6 meters per second per second. A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?64E65E66E67E68EAcceleration The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assume the acceleration is constant. (a) Find the acceleration in meters per second per second. (b) Find the distance the car travels during the 13 seconds.Deceleration A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied. (a) How far has the car moved when its speed has been reduced to 30 miles per hour? (b) How far has the car moved when its speed has been reduced to 15 miles per hour? (c) Draw the real number line from 0 to 132. Plot the points found in parts (a) and (b). What can you conclude?71E72ETrue or False? In Exercises 73 and 74, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The antiderivative of f(x) is unique.74E79E80ETrue or False? In Exercises 73-78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If p(x) is a polynomial function, then p has exactly one antiderivative whose graph contains the origin.76E77E81E78ECONCEPT CHECK Sigma Notation What are the index of summation, the upper bound of summation, and the lower bound of summation for i=38(i4)?2E3E4EFinding a Sum In Exercises 5-10, find the sum by adding each term together. Use the summation capabilities of a graphing utility to verify your result i=16(3i+2)6EFinding a Sum In Exercises 5-10, find the sum by adding each term together. Use the summation capabilities of a graphing utility to verify your result k=041k2+18EFinding a Sum In Exercises 5-10, find the sum by adding each term together. Use the summation capabilities of a graphing utility to verify your result k=07c10E11E12E13E14E15E16E17E18EEvaluating a Sum In Exercises 17-24, use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of graphing utility to verify your result. i=1244i20E21E22E23EEvaluating a Sum In Exercises 1724, use the properties of summation and Theorem 5.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. i=125(i32i)25E26E27EEvaluating a Sum In Exercises 25-28, use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for n=10,100,1000, and 10,000. i=1n2i33in429EApproximating the Area of a Plane Region In Exercises 29-34, use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. f(x)=9x,[2,4],6rectangles31E32E33E34E35EUsing Upper and Lower Sums In Exercises 35 and 36, bound the area of the shaded region by approximating the upper and lower sums. Use rectangles of width 1.37EFinding Upper and Lower Sums for a Region In Exercises 37-40, use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). y=x+2Finding Upper and Lower Sums for a Region In Exercises 37-40, use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). y=1x40EFinding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Function Interval f(x)=3x [0,4]Finding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Function Interval f(x)=62x 1,2Finding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Function Interval f(x)=5x2 0,1Finding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals. Function Interval f(x)=9x2 [0,2]Numerical Reasoning Consider a triangle of area 2 bounded by the graphs of y=x,y=0, and x=2. (a) Sketch the region. (b) Divide the interval [0, 2] into n subintervals of equal width and show that the endpoints are 01(2n)(n1)(2n)n(2n) (c) Show that s(n)=i=1n[(i1)(2n)](2n) (d) show that S(n)=i=1n[i(2n)](2n) (e) Find s(n) and S(n) for n=5,10,50, and 100. (f) Show that limns(n)=limnS(n)=2Numerical Reasoning Consider a triangle of area 4 bounded by the graphs of y=x,y=0,x=1, and x=3. (a) Sketch the region. (b) Divide the interval [1, 3] into n subintervals of equal width and show that the endpoints are 11+1(2n)1+(n1)(2n)1+n(2n) (c) Show that s(n)=i=1n[ 1+(i1)(2n) ](2n) (d) show that S(n)=i=1n[ 1+i(2n) ](2n) (e) Find s(n) and S(n) for n=5,10,50, and 100. (f) Show that limns(n)=limnS(n)=447E48E49E50E51E52E