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All Textbook Solutions for Calculus: Early Transcendentals
For what values of the constants a and b is (1,3) a point of inflection of the curve y = ax3 + bx2?49REFind two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.Show that the shortest distance from the point (x1, y1) to the straight line Ax + By + C = 0 is |Ax1+By1+C|A2+B2Find the point on the hyperbola xy = 8 that is closest to the point (3, 0).Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.54RE55RE56REThe velocity of a wave of length L in deep water is v=KLC+CL where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at 12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?A manufacturer determines that the cost of making x units of a commodity is C(x)=1800+25x0.2x2+0.001x3 and the demand function is p(x) = 48.2 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.61RE62RE63RE64RE65RE66REFind the most general antiderivative of the function. f(t) = 2 sin t 3et68REFind f. f(t) = 2t 3 sin t, f(0) = 570RE71RE72RE73RE74RE75RE76REA canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.A rectangular beam will be cut from a cylindrical log of radius 10 inches. (a) Show that the beam of maximal cross-sectional area is a square. (b) Four rectangular planks will be cut from the four sections of the log that remain after cutting the square beam. Determine the dimensions of the planks that will have maximal cross-sectional area. (c) Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.If a projectile is fired with an initial velocity v at an angle of inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola y=(tan)xg2v2cos2x202 (a) Suppose the projectile is fired from the base of a plane that is inclined at an angle , 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by R()=2v2cossin()gcos2 (b) Determine so that R is a maximum, (c) Suppose the plane is at an angle below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is P(E)=eE+eEeEeE1E Show that limE0+P(E)=0.If a metal ball with mass m is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the hall travels in time t is s(t)=mclncoshgcmt where c is a positive constant. Find limc0+s(t).83RE84RE85REWater is flowing at a constant rate into a spherical tank. Let V(t) be the volume of water in the tank and H(t) be the height of the water in the tank at time t. (a) What are the meanings of V(t) and H(t)? Are these derivatives positive, negative, or zero? (b) Is V(t) positive, negative, or zero? Explain. (c) Let t1 t2, and t3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H(t1), H(t2), and H(t3) positive, negative, or zero? Why?1P2P3P4PShow that the inflection points of the curve y = (sin x)/x lie on the curve y2(x4 + 4) = 4.Find the point on the parabola y = 1 x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.If a, b, c, and d are constants such that limxax2+sinbx+sincx+sindx3x2+5x4+7x6=8 find the value of the sum a + b + c + d.8PFind the highest and lowest points on the curve x2 + xy + y2 = 12.10PIf P(a, a2) is any point on the parabola y = x2, except for the origin, let Q be the point where the normal line at P intersects the parabola again (see the figure). (a) Show that the y-coordinate of Q is smallest when a=1/2. (b) Show that the line segment PQ has the shortest possible length when a=1/2.For what values of c does the curve y = cx3 + ex have inflection points?An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point (0, a) on the y-axis (see the figure). Find the value of a that minimizes the lengths of the equal sides. (You may be surprised that the result does not give an equilateral triangle.).14PThe line y = mx + b intersects the parabola y = x2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB.ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas that the triangle AEF can have.For which positive numbers a does the curve y = ax intersect the line y = x?18PLet f(x) = a1 sin x + a2 sin 2x + + an, sin nx, where a1, a2, , an are real numbers and n is a positive integer. If it is given that | f(x) | |sin x| for all x, show that |a1 + 2a2 + + nan| 1An arc PQ of a circle subtends a central angle as in the figure. Let A() be the area between the chord PQ and the arc PQ. Let B() be the area between the tangent lines PR, QR, and the arc. Find lim0+A()B()The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at a point P and the transmitted signals are recorded at a point Q, which is a distance D from P. The first signal to arrive at Q travels along the surface and takes T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (See the figure.) (a) Express T1, T2, and T3 in terms of D, h, c1, c2, and . (b) Show that T2, is a minimum when sin = c1/c2. (c) Suppose that D = 1 km, T1, = 0.26 s, T2 = 0.32 s, and T3 = 0.34 s. Find c1, c2, and h. Note: Geophysicists use this technique when studying the structure of the earths crust, whether searching for oil or examining fault lines.For what values of c is there a straight line that intersects the curve y=x4+cx3+12x25x+2 in four distinct points?One of the problems posed by the Marquis de IHospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d r), a rope of length is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. (See the figure.) As IHospital argued, this happens when the distance | ED | is maximized. Show that when the system reaches equilibrium, the value of x is r4d(r+r2+8d2)25PA hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1+n.1E(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12. (i) L6 (sample points are left endpoints) (ii) R6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L6 an underestimate or overestimate of the true area? (c) Is R6 an underestimate or overestimate of the true area? (d) Which of the numbers L6, R6, or M6 gives the best estimate? Explain.(a) Estimate the area under the graph of f(x) = 1/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.4E(a) Estimate the area under the graph of f(x) = 1 + x2 from x = 1 to x = 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)(c), which appears to be the best estimate?6E7E8EThe speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.The table shows speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida. (a) Estimate the distance the race car traveled during this time period using the velocities at the beginning of the time intervals. Time (s) Velocity (mi/h) 0 182.9 10 168.0 20 106.6 30 99.8 40 124.5 50 176.1 60 175.6 (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.When we estimate distances from velocity data, it is sometimes necessary to use times t0, t1, t2, t3, that are not equally spaced. We can still estimate distances using the time periods ti = ti ti 1. For example, 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, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the earths surface of the Endeavour, 62 seconds after liftoff. 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 4151The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.In someone infected with measles, the virus level N (measured in number of infected cells per mL of blood plasma) reaches a peak density at about t = 12 days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of N(t) from t = 0 to t = 12 (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells time). The function N has been modeled by the function f(t)=t(t21)(t+1) Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles. Source: J. M. Heffernan et al., An In-Host Model of Acute Infection: Measles as a Case Study, Theoretical Population Biology 73 (2006): 13447.The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003. Date Deaths per day March 1 0.0079 March 15 0.0638 March 29 0.1944 April 12 0.4435 April 26 0.5620 May 10 0.4630 May 24 0.2897 (a) By using an argument similar to that in Example 4, estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. (b) How would you interpret the number of SARS deaths as an area under a curve? Source: A. Gumel et al., Modelling Strategies for Controlling SARS Outbreaks, Proceedings of the Royal Society of London: Series B 271 (2004): 222332.Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=2xx2+1,1x3Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=x2+1+2x,4x7Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=sinx,0x24E25E(a) Use Definition 2 to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13+23+33++n3=[n(n+1)2]2Let A be the area under the graph of an increasing continuous function f from a to b, and let Ln and Rn be the approximations to A with n subintervals using left and right endpoints, respectively. (a) How are A, Ln, and Rn related? (b) Show that RnLn=ban[f(b)f(a)] Then draw a diagram to illustrate this equation by showing that the n rectangles representing Rn Ln can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that RnAban[f(b)f(a)]If A is the area under the curve y = ex from 1 to 3, use Exercise 27 to find a value of n such that Rn A 0.0001.(a) Let An be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2/n, show that An=12nr2sin(2n) (b) Show that limn An = r2. [Hint: Use Equation 3.3.2 on page 191.]Evaluate the Riemann sum for f(x) = x 1, 6 x 4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.If f(x)=cosx0x3/4 evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.If f(x) = x2 4, 0 x 3, find the Riemann sum with n = 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.(a) Find the Riemann sum for f(x) = 1/x, 1 x 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points.The graph of a function f is given. Estimate 010f(x)dxusing five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints.The graph of g is shown. Estimate 24g(x)dx with six subintervals using (a) right endpoints, (b) left endpoints, and (c) midpoints.A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for 1030f(x)dx.The table gives the values of a function obtained from an experiment. Use them to estimate 39f(x)dx using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral?Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 08sinxdx,n=4Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 01x3+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 02xx+1dx,n=5Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 0xsin2xdx,n=4With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f(x) = x/(x + 1) on the interval [0, 2] with n = 100. Explain why these estimates show that 0.894602xx+1dx0.908115EUse a calculator or computer to make a table of values of left and right Riemann sums Ln and Rn for the integral 02ex2dx with n = 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a similar statement for the integral 12ex2dx? Explain.Express the limit as a definite integral on the given interval. limni=1nexi1+xix,[0,1]Express the limit as a definite integral on the given interval. limni=1nxi1+xi3x,[2,5]Express the limit as a definite integral on the given interval. limni=1n[5(xi)34xi]x,[2,7]Express the limit as a definite integral on the given interval. limni=1nxi(xi)2+4x,[1,3]Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 25(42x)dxUse the form of the definition of the integral given in Theorem 4 to evaluate the integral. 14(x24x+2)dxUse the form of the definition of the integral given in Theorem 4 to evaluate the integral. 20(x2+x)dxUse the form of the definition of the integral given in Theorem 4 to evaluate the integral. 02(2xx3)dxUse the form of the definition of the integral given in Theorem 4 to evaluate the integral. 01(x33x2)dx(a) Find an approximation to the integral 04(x23x)dx using a Riemann sum with right endpoints and n = 8. (b) Draw a diagram like Figure 3 to illustrate the approximation in part (a). (c) Use Theorem 4 to evaluate 04(x23x)dx. (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4.Prove that abxdx=b2a22.Prove that abx2dx=b3a33.Express the integral as a limit of Riemann sums. Do not evaluate the limit. 134+x2dxExpress the integral as a limit of Riemann sums. Do not evaluate the limit. 25(x2+1x)dxThe graph of f is shown. Evaluate each integral by interpreting it in terms of areas. (a) 02f(x)dx (b) 05f(x)dx (c) 57f(x)dx (d) 09f(x)dxThe graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) 02g(x)dx (b) 26g(x)dx (c) 07g(x)dxEvaluate the integral by interpreting it in terms of areas. 12(1x)dxEvaluate the integral by interpreting it in terms of areas. 09(13x2)dxEvaluate the integral by interpreting it in terms of areas. 30(1+9x2)dxEvaluate the integral by interpreting it in terms of areas. 55(x=25x2)dxEvaluate the integral by interpreting it in terms of areas. 4312xdxEvaluate the integral by interpreting it in terms of areas. 012x1dxEvaluate 111+x4dx.Given that 0sin4xdx=83, what is 0sin4d?In Example 5.1.2 we showed that 01x2dx13. Use this fact and the properties of integrals to evaluate 01(56x2)dx.Use the properties of integrals and the result of Example 3 to evaluate 13(2ex1)dx.Use the result of Example 3 to evaluate 13ex+2dx.46EWrite as a single integral in the form abf(x)dx: 22f(x)dx+25f(x)dx21f(x)dxIf 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.If 09f(x)dx=37 and 09g(x)dx=16, find 09[2f(x)+3g(x)]dxFind 05f(x)dx if f(x)={3forx3xforx3For the function f whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning. (A) 08f(x)dx (B)03f(x)dx (C)38f(x)dx (D)48f(x)dx (E) f(1)If , F(x)=2xf(t)dt, where f is the function whose graph is given, which of the following values is largest? (A) F(0) (B) F(1) (C) F(2) (D) F(3) (E) F(4)Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of 42[f(x)+2x+5]dxSuppose f has absolute minimum value m and absolute maximum value M. Between what two values must 02f(x)dx lie? Which property of integrals allows you to make your conclusion?Use the properties of integrals to verify the inequality without evaluating the integrals. 04(x24x+4)dx0Use the properties of integrals to verify the inequality without evaluating the integrals. 011+x2dx011+xdxUse the properties of integrals to verify the inequality without evaluating the integrals. 2111+x2dx22Use the properties of integrals to verify the inequality without evaluating the integrals. 12/6/3sinxdx312Use Property 8 to estimate the value of the integral. 01x3dxUse Property 8 to estimate the value of the integral. 031x+4dxUse Property 8 to estimate the value of the integral. /4/3tanxdxUse Property 8 to estimate the value of the integral. 02(x33x+3)dxUse Property 8 to estimate the value of the integral. 02xexdxUse Property 8 to estimate the value of the integral. 2(x2sinx)dxUse properties of integrals, together with Exercises 27 and 28, to prove the inequality. 13x4+1dx263Use properties of integrals, together with Exercises 27 and 28, to prove the inequality. 0/2xsinxdx28Which of the integrals 12arctanxdx, 12arctanxdx, and 12arctan(sinx)dx has the largest value? Why?Which of the integrals 00.5cos(x2)dx, 00.5cosxdx is larger? Why?69E(a) If f is continuous on [a, b], show that |abf(x)dx|abf(x)dx [Hint:| f(x) | f(x) | f(x) |.] (b) Use the result of part (a) to show that |02f(x)sin2xdx|02f(x)dxLet f(x) = 0 if x is any rational number and f(x) = 1 if x is any irrational number. Show that f is not integrable on [0, 1].Let f(0) = 0 and f(x) = 1/x if 0 x 1. Show that f is not integrable on [0, 1]. [Hint: Show that the first term in the Riemann sum, f(x1)x, can be made arbitrarily large.]Express the limit as a definite integral. limni=1ni4n5 [Hint: Consider f(x) = x4.]Express the limit as a definite integral. limn1ni=1n11+(i/n)2Find 12x2dx. Hint: Choose xi to be the geometric mean of xi1 and xi (thatis,xi=xi1xi) and use the identity 1m(m+1)=1m1m+1Explain exactly what is meant by the statement that differentiation and integration are inverse processes.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, and 6. (b) Estimate g(7). (c) Where does g have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of g.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) Evaluate g(0), g(1), g(2), g(3), and g(6). (b) On what interval is g increasing? (c) Where does g have a maximum value? (d) Sketch a rough graph of g.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) Evaluate g(0) and g(6). (b) Estimate g(x) for x = 1, 2, 3, 4, and 5. (c) On what interval is g increasing? (d) Where does g have a maximum value? (e) Sketch a rough graph of g. (f) Use the graph in part (e) to sketch the graph of g(x). Compare with the graph of f.Sketch the area represented by g(x). Then find g(x) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. g(x)=1xt2dtSketch the area represented by g(x). Then find g(x) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. g(x)=0x(2+sint)dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(x)=0xt+t3dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(x)=1xln(1+t2)dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(s)=5s(tt2)8dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(u)=0utt+1dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x)=x01+sectdt [Hint:x01+sectdt=0x1+sectdt]Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. R(y)=y2t3sintdtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=1exlntdtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=1xz2z4+1dzUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=13x+2t1+t3dtUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=0x4cos2dUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=x/4tandUse Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=sinx11+t2dtEvaluate the integral. 13(x2+2x4)dxEvaluate the integral. 11x100dxEvaluate the integral. 02(45t334t2+25t)dtEvaluate the integral. 01(18v3+16v7)dvEvaluate the integral. 19xdxEvaluate the integral. 18x2/3dxEvaluate the integral. /6sindEvaluate the integral. 55edxEvaluate the integral. 01(u+2)(u3)duEvaluate the integral. 04(4t)tdtEvaluate the integral. 142+x2xdxEvaluate the integral. 12(3u2)(u+1)duEvaluate the integral. /6/2csctcottdtEvaluate the integral. /4/3csc2dEvaluate the integral. 01(1+r)3drEvaluate the integral. 03(2sinxex)dxEvaluate the integral. 12v3+3v6v4dvEvaluate the integral. 1183zdzEvaluate the integral. 01(xe+ex)dxEvaluate the integral. 01coshtdtEvaluate the integral. 1/3381+x2dxEvaluate the integral. 13y32y2yy2dyEvaluate the integral. 042sdsEvaluate the integral. 1/21/241x2dxEvaluate the integral. 0f(x)dxwheref(x)={sinxif0x/2cosxif/2xEvaluate the integral. 22f(x)dxwheref(x)={2if2x04x2if0x2Sketch the region enclosed by the given curves and calculate its area. y=x,y=0,x=4Sketch the region enclosed by the given curves and calculate its area. y = x3, y = 0, x = 1Sketch the region enclosed by the given curves and calculate its area. y = 4 x2, y = 0Sketch the region enclosed by the given curves and calculate its area. y = 2x x2, y = 0Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y=x3,0x2750EUse a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = sin x, 0 xUse a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = sec2x, 0 x /3Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. 12x3dx54EWhat is wrong with the equation? 21x4dx=x33]21=3856EWhat is wrong with the equation? /3sectand=sec]/3=3What is wrong with the equation? 0sec2xdx=tanx]0=0Find the derivative of the function. g(x)=2x3xu21u2+1du [Hint:2x3xf(u)du=2x0f(u)du+03xf(u)du]Find the derivative of the function. g(x)=12x1+2xtsintdtFind the derivative of the function. F(x)=xx2et2dtFind the derivative of the function. F(x)=x2xarctantdtFind the derivative of the function. y=cosxsinxln(1+2v)dvIf f(x)=0x(1t2)et2dt, on what interval is f increasing?On what interval is the curve y=0xt2t2+t+2dt concave downward?Let F(x)=1xf(t)dt, where f is the function whose graph is shown. Where is F concave downward?Let F(x)=2xet2dt. Find an equation of the tangent line to the curve y = F(x) at the point with x-coordinate 2.If f(x)=0sinx1+t2dt and g(y)=3yf(x)dx, find g(/6).69EThe error function erf(x)=20xet2dt is used in probability, statistics, and engineering. (a) Show that abet2dt=12[erf(b)erf(a)]. (b) Show that the function y=ex2erf(x)satisfies the differential equation y=2xy+2/.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) At what values of x do the local maximum and minimum values of g occur? (b) Where does g attain its absolute maximum value? (c) On what intervals is g concave downward? (d) Sketch the graph of g.Let g(x)=0xf(t)dt, where f is the function whose graph is shown. (a) At what values of x do the local maximum and minimum values of g occur? (b) Where does g attain its absolute maximum value? (c) On what intervals is g concave downward? (d) Sketch the graph of g.Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. limni=1n(i4n5+in2)Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. limn1n(1n+2n+3n++nn)77EIf f is continuous and g and h are differentiable functions, find a formula for ddxg(x)h(x)f(t)dt(a) Show that 11+x31+x3 for x 0. (b) Show that 1011+x3dx1.25.(a) Show that cos(x2) cos x for 0 x 1. (b) Deduce that 0/6cos(x2)dx12.Show that 0510x2x4+x2+1dx0.1 by comparing the integrand to a simpler function.Let f(x)={0ifx0xif0x12xif1x20ifx2 and g(x)=0xf(t)dt (a) Find an expression for g(x) similar to the one for f(x). (b) Sketch the graphs of f and g. (c) Where is f differentiable? Where is g differentiable?Find a function f and a number a such that 6+axf(t)t2dt=2xforallx0The area labeled B is three times the area labeled A. Express b in terms of a.A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate f = f(t), where t is the time measured in months since its last overhaul. Because a fixed cost A is incurred each time the machine is overhauled, the company wants to determine the optimal time T (in months) between overhauls. (a) Explain why 0tf(s)ds represents the loss in value of the machine over the period of time t since the last overhaul. (b) Let C = C(t) be given by C(t)=1t[A+0tf(s)ds] What does C represent and why would the company want to minimize C? (c) Show that C has a minimum value at the numbers t = T where C(T) = f(T).A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f = f(t) and will accumulate maintenance costs at the rate g = g(t), where t is the time measured in months. The company wants to determine the optimal time to replace the system. (a) Let C(t)=1t0t[f(s)+g(s)]ds Show that the critical numbers of C occur at the numbers t where C(t) = f(t) + g(t). (b) Suppose that f(t)={V15V450tif0t300ift30 and g(t)Vt212,900t0 Determine the length of time T for the total depreciation D(t)=0tf(s)ds to equal the initial value V. (c) Determine the absolute minimum of C on (0, T]. (d) Sketch the graphs of C and f + g in the same coordinate system, and verify the result in part (a) in this case.Verify by differentiation that the formula is correct. 1x21+x2dx=1+x2x+CVerify by differentiation that the formula is correct. cos2xdx=12x+14sin2x+CVerify by differentiation that the formula is correct. tan2xdx=tanxx+CVerify by differentiation that the formula is correct. xa+bxdx=215b2(3bx2a)(a+bx)3/2+CFind the general indefinite integral. (x1.3+7x2.5)dxFind the general indefinite integral. x54dxFind the general indefinite integral. (5+23x2+34x3)dxFind the general indefinite integral. (u62u5u3+27)duFind the general indefinite integral. (u+4)(2u+1)duFind the general indefinite integral. t(t2+3t+2)dtFind the general indefinite integral. 1+x+xxdxFind the general indefinite integral. (x2+1+1x2+1)dxFind the general indefinite integral. (sinx+sinhx)dx14EFind the general indefinite integral. (2+tan2)d16E17EFind the general indefinite integral. sin2xsinxdxFind the general indefinite integral. Illustrate by graphing several members of the family on the same screen. (cosx+12x)dxFind the general indefinite integral. Illustrate by graphing several members of the family on the same screen. (ex2x2)dxEvaluate the integral. 23(x23)dxEvaluate the integral. 12(4x33x2+2x)dxEvaluate the integral. 20(12t4+14t3t)dtEvaluate the integral. 03(1+6w210w4)dwEvaluate the integral. 02(2x3)(4x2+1)dxEvaluate the integral. 11t(1t)2dtEvaluate the integral. 0(5ex+3sinx)dxEvaluate the integral. 12(1x24x3)dxEvaluate the integral. 14(4+6uu)duEvaluate the integral. 0141+p2dpEvaluate the integral. 01x(x3+x4)dx32EEvaluate the integral. 12(x22x)dxEvaluate the integral. 01(5x5x)dxEvaluate the integral. 01(x10+10x)dxEvaluate the integral. 0/4sectandEvaluate the integral. 0/41+cos2cos2dEvaluate the integral. 0/3sin+sintan2sec2dEvaluate the integral. 182+tt23dtEvaluate the integral. 10102exsinhx+coshxdxEvaluate the integral. 03/2dr1r2Evaluate the integral. 02(x1)3x2dxEvaluate the integral. 01/3t21t41dtEvaluate the integral. 022x1dxEvaluate the integral. 12(x2x)dxEvaluate the integral. 03/2sinxdx47E48EThe area of the region that lies to the right of the y-axis and to the left of the parabola x = 2y y2 (the shaded region in the figure) is given by the integral 02(2yy2)dy. (Turn your head clockwise and think of the region as lying below the curve x = 2y y2 from y = 0 to y = 2.) Find the area of the region.The boundaries of the shaded region are the y-axis, the line y = 1, and the curve y=x4. Find the area of this region by writing x as a function of y and integrating with respect to y (as in Exercise 49).If w(t) is the rate of growth of a child in pounds per year, what does 510w(t)dt represent?52EIf oil leaks from a tank at a rate of r(t) gallons per minute at time t, what does 0120r(t)dt represent?A honeybee population starts with 100 bees and increases at a rate of n(t) bees per week. What does 100+015n(t)dt represent?In Section 4.7 we defined the marginal revenue function R(x) as the derivative of the revenue function R(x), where x is the number of units sold. What does 10005000R(x)dx represent?