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All Textbook Solutions for Calculus Volume 1
For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 391. [T] tanh(x2+1)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 392. [T] 1+tanh(x)1tanh(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 393. [T] sinh6(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 394. [T] In(sech(x)+tanh(x))For the following exercises, find the antiderivatives for the given functions. 395. cosh(2x+1)For the following exercises, find the antiderivatives for the given functions. 396. tanh(3x+2)For the following exercises, find the antiderivatives for the given functions. 397. xcosh(x2)For the following exercises, find the antiderivatives for the given functions. 398. 3x3tanh(x4)For the following exercises, find the antiderivatives for the given functions. 399. cosh2(x)sinh(x)For the following exercises, find the antiderivatives for the given functions. 400. tan2(x)sech2(x)For the following exercises, find the antiderivatives for the given functions. 401. sinh(x)1+cosh(x)For the following exercises, find the antiderivatives for the given functions. 402. coth(x)For the following exercises, find the antiderivatives for the given functions. 403. cosh(x)+sinh(x)For the following exercises, find the antiderivatives for the given functions. 404. (cosh(x)+sinh(x))nFor the following exercises, find the derivatives for the functions. 405. tanh1(4x)For the following exercises, find the derivatives for the functions. 406. sinh1(x2)For the following exercises, find the derivatives for the functions. 407. sinh1(cosh(x))For the following exercises, find the derivatives for the functions. 408. cosh1(x3)For the following exercises, find the derivatives for the functions. 409. tanh1(cos(x))For the following exercises, find the derivatives for the functions. 410. esinh1(x)For the following exercises, find the derivatives for the functions. 411. In(tanh1(x))For the following exercises, find the antiderivatives for the functions. 412. dx4 x 2For the following exercises, find the antiderivatives for the functions. 413. dx a 2 x 2For the following exercises, find the antiderivatives for the functions. 414. dx x 2 1 .For the following exercises, find the antiderivatives for the functions. 415. xdx x 2 1For the following exercises., find the antiderivatives for the functions. 416. dxx 1 x 2For the following exercises., find the antiderivatives for the functions. 417. e x e 2x 1For the following exercises, find the antiderivatives for the functions. 418. 2x x 41For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt=gv2. 419. Show that v(t)=gtanh(gt) satisfies this equation.For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt=gv2. 420. Derive the previous expression for v(t) by integrating dvgv2=dt.For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation dv/dt=gv2. 421. [T] Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of v(t) .For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2 . The constant c is the ratio of cable density to tension. 422. S=sinh(cx)For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2 . The constant c is the ratio of cable density to tension. 423. Integrate dy/dx=sinh(cx) to find the cable height y(x) if y(0)=1/c .For the following exercises, use this scenario: A cable hanging under its own weight has a slope S=dy/dx that satisfies dS/dx=c1+S2 . The constant c is the ratio of cable density to tension. 424. Sketch the cable and determine how far down it sags at x=0 .For the following exercises, solve each problem. 425. [T] A chain hangs from two posts 2 m apart to form a catenary described by tine equation y=2cosh(x/2)1 .Find the slope of the catenary at the left fence post.For the following exercises, solve each problem. 426. [T] A chain hangs from two posts four meters apart to form a catenary described by the equation y=4cosh(x/4)3 . Find the total length of the catenary (arc length).For the following exercises, solve each problem. 427. [T] A high-voltage power line is a catenary described by y=10cosh(x/10) . Find the ratio of the area tinder the catenary to its arc length. What do you notice?For the following exercises, solve each problem. 428. A telephone line is a catenary described by y=acosh(x/a) . Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?For the following exercises, solve each problem. 429. Prove the formula for the derivative of y=sinh1(x) by differentiating x=sinh(y) . (Hint: Use hyperbolic trigonometric identities.)For the following exercises, solve each problem. 430. Prove the formula for the derivative of y=cosh1(x) by differentiating x=cosh(y) . (Hint: Use hyperbolic trigonometric identities.)For the following exercises, solve each problem. 431. Prove the formula for the derivative of y=sech1(x) by differentiating x=sech(y) . (Hint: Use hyperbolic trigonometric identities.)Prove that (cosh(x)+sinh(x)n)=cosh(nx)+sinh(nx) .Prove the expression for sinh1(x) . Multiply x=sinh(y)=(1/2)(eyey) by 2ey and solve for y. Does your expression match the textbook?Prove the expression for cosh1(x) . Multiply x=cosh(y)=(1/2)(eyey) by 2ey and solve for y. Does your expression match die textbook?True or False ? Justify your answer with a proof or a counterexample. 435. The amount of work to pump the water out of a halffull cylinder is half the amount of work to pump the water out of the full cylinder.True or False ? Justify your answer with a proof or a counterexample. 436. If the force is constant, the amount of work to move an object from x=a to x=b is F(ba).The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.If the half-life of seaborsiuni-266 is 360 ms, then k=(In(2)/360) .For the following exercises, use the requested method to determine the volume of the solid. 439. The volume that has a base of the ellipse x2/9+y2/9=1 and cross-sections of an equilateral triangle perpendicular to the y-axis. Use the method of slicing.For the following exercises, use the requested method to determine the volume of the solid. 440. y=x2x, from x=1to x=4, rotated around the y-axis using the washer methodFor the following exercises, use the requested method to determine the volume of the solid. 441. x=y2 and x=3y rotated around the y-axis using the washer methodFor the following exercises, use the requested method to determine the volume of the solid. 442. x=2y2y3,x=0 , and y=0 rotated around the x-axis using cylindrical shellsFor the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 443. y=x3,x=0,y=0 and x=2For the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 444. y=x2x and x=0For the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 445. [T] y=In(x)+2 and y=xFor the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 446. y=x2 and y=xFor the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 447. y=5+x,y=x2,x=0 , and x=1For the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 448. Below x2+y2=1 and above y=1xFor the following exercises, find the area of the region, the volume of the solid when rotated around the x-axis, and the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 449. Find the mass of =ex on a disk centered at the origin with radius 4 .Find the center of mass for =tan2x on x(4,4) .Find the mass and the center of mass of =1 on the region bounded by y=x5 and y=x .For the following exercises, find the requested arc lengths. 452. The length of x for y=cosh(x) from x=0 to x=2 .For the following exercises, find the requested arc lengths 453. The length of y for x=3y from y=0 to y=4For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis. 454. The shape created by revolving the region between y=4+x,y=3x,x=0 , and x=2 rotated around the y-axis.For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis. 455. The loudspeaker created by revolving y=1/x from x=1 to x=4 around the x-axis.For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 m and width 388 m. Assume the current depth of the water is 180 m. The density of water is 1000kg/m3 . 456. Find the total force on the wall of the damYou are a crime scene investigator attempting to determine the time of death of a victim. It is noon and 45°F outside and the temperature of the body is 78°F. You know the cooling constant is k = 0.00824°F/min. When did the victim die, assuming that a human’s temperature is 98°F ?For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash. Years after 1920 Value($) 1 63.90 3 100 5 110 7 160 9 381.17 Source: http://stockcharts.com/freecharts/historical/ djia19201940.html 458. [T] The best-fit exponential curve to these data is given by y=40.71+1.224x . Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014 ?For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings. 459. Find the volume of the catenoid y=cosh(x) from x=1 to x=1 that is created by rotating this curve around the x-axis, as shown here.Find surface area of the catenoid y=cosh(x) from x=1 to x=1 that is created by rotating this curve around the x-axis.