Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term
8th Edition
ISBN: 9781305597624
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 6.4, Problem 23E
A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.
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MATH 122
WORKSHEET 3
February 5, 2025
. Solve the following problems on a separate sheet. Justify your answers to earn full credit.
1. Let f(x) = x² - 2x + 1.
(a) Find the slope of the graph of y = f (x) at the point P = (0,1) by directly
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f'(0) = lim (
f(Ax) - f(0)
Ax
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(b) Find the equation of the tangent line 1 to the graph of ƒ at P.
What are the x and y intercepts of 1 ?
(c) Find the equation of the line, n, through P that is perpendicular to the tangent line l.
(Line n is called the normal line to the graph of f at P.)
(d) Sketch a careful graph that displays: the graph of y = f (x), its vertex point, its
tangent and normal lines at point P, and the x and y intercepts of these lines.
Bonus: Find the coordinates of the second point, Q, (QP), at which the normal
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from the edge of a bridge that is 25 meters above a river bed. Based
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Chapter 6 Solutions
Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term
Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....
Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Prob. 29ECh. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Sketch the region enclosed by the given curves and...Ch. 6.1 - Use calculus to find the area of the triangle with...Ch. 6.1 - Use calculus to find the area of the triangle with...Ch. 6.1 - Evaluate the integral and interpret it as the area...Ch. 6.1 - Evaluate the integral and interpret it as the area...Ch. 6.1 - Use a graph to find approximate x-coordinates of...Ch. 6.1 - Prob. 38ECh. 6.1 - Prob. 39ECh. 6.1 - Prob. 40ECh. 6.1 - Graph the region between the curves and use your...Ch. 6.1 - Prob. 42ECh. 6.1 - Prob. 43ECh. 6.1 - Prob. 44ECh. 6.1 - Sketch the region in the xy-plane defined by the...Ch. 6.1 - Racing cars driven by Chris and Kelly are side by...Ch. 6.1 - The widths (in meters) of a kidney-shaped swimming...Ch. 6.1 - A cross-section of an airplane wing is shown....Ch. 6.1 - If the birth rate of a population is b(t) =...Ch. 6.1 - In Example 5, we modeled a measles pathogenesis...Ch. 6.1 - Prob. 52ECh. 6.1 - Two cars, A and B, start side by side and...Ch. 6.1 - The figure shows graphs of the marginal revenue...Ch. 6.1 - The curve with equation y2 = x2(x + 3) is called...Ch. 6.1 - Find the area of the region bounded by the...Ch. 6.1 - Find the number b such that the line y = b divides...Ch. 6.1 - (a) Find the number a such that the line x = a...Ch. 6.1 - Find the values of c such that the area of the...Ch. 6.1 - Suppose that 0 c /2. For what value of c is the...Ch. 6.1 - For what values of m do the line y = mx and the...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Refer to the figure and find the volume generated...Ch. 6.2 - Set up an integral for the volume of the solid...Ch. 6.2 - Set up an integral for the volume of the solid...Ch. 6.2 - Set up an integral for the volume of the solid...Ch. 6.2 - Set up an integral for the volume of the solid...Ch. 6.2 - Use a graph to find approximate x-coordinates of...Ch. 6.2 - Use a graph to find approximate x-coordinates of...Ch. 6.2 - Each integral represents the volume of a solid....Ch. 6.2 - Each integral represents the volume of a solid....Ch. 6.2 - Each integral represents the volume of a solid....Ch. 6.2 - Each integral represents the volume of a solid....Ch. 6.2 - A CAT scan produces equally spaced cross-sectional...Ch. 6.2 - A log 10 m long is cut at 1-meter intervals and...Ch. 6.2 - (a) If the region shown in the figure is rotated...Ch. 6.2 - Find the volume of the described solid S. A right...Ch. 6.2 - Find the volume of the described solid S. A...Ch. 6.2 - Find the volume of the described solid S. A cap of...Ch. 6.2 - Find the volume of the described solid S. A...Ch. 6.2 - Find the volume of the described solid S. A...Ch. 6.2 - Find the volume of the described solid S. A...Ch. 6.2 - Find the volume of the described solid S. A...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The base...Ch. 6.2 - Find the volume of the described solid S. The...Ch. 6.2 - The base of S is a circular disk with radius r....Ch. 6.2 - (a) Set up an integral for the volume of a solid...Ch. 6.2 - Prob. 64ECh. 6.2 - (a) Cavalieris Principle states that if a family...Ch. 6.2 - Find the volume common to two circular cylinders,...Ch. 6.2 - Prob. 67ECh. 6.2 - A bowl is shaped like a hemisphere with diameter...Ch. 6.2 - A hole of radius r is bored through the middle of...Ch. 6.2 - A hole of radius r is bored through the center of...Ch. 6.2 - Some of the pioneers of calculus, such as Kepler...Ch. 6.2 - Suppose that a region has area A and lies above...Ch. 6.3 - Let S be the solid obtained by rotating the region...Ch. 6.3 - Let S be the solid obtained by rotating the region...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Let V be the volume of the solid obtained by...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - Use the method of cylindrical shells to find the...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - (a) Set up an integral for the volume of the solid...Ch. 6.3 - Use the Midpoint Rule with n = 5 to estimate the...Ch. 6.3 - If the region shown in the figure is rotated about...Ch. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Use a graph to estimate the x-coordinates of the...Ch. 6.3 - Prob. 34ECh. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - The region bounded by the given curves is rotated...Ch. 6.3 - Let T be the triangular region with vertices (0,...Ch. 6.3 - Prob. 45ECh. 6.3 - Prob. 46ECh. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.4 - A 360-lb gorilla climbs a tree to a height of 20...Ch. 6.4 - How much work is done when a hoist lifts a 200-kg...Ch. 6.4 - Prob. 3ECh. 6.4 - When a particle is located a distance x meters...Ch. 6.4 - Shown is the graph of a force function (in...Ch. 6.4 - Prob. 6ECh. 6.4 - A force of 10 lb is required to hold a spring...Ch. 6.4 - A spring has a natural length of 40 cm. If a 60-N...Ch. 6.4 - Suppose that 2 J of work is needed to stretch a...Ch. 6.4 - If the work required to stretch a spring 1 ft...Ch. 6.4 - A spring has natural length 20 cm. Compare the...Ch. 6.4 - If 6 J of work is needed to stretch a spring from...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - Show how to approximate the required work by a...Ch. 6.4 - A tank is full of water. Find the work required to...Ch. 6.4 - A tank is full of water. Find the work required to...Ch. 6.4 - A tank is full of water. Find the work required to...Ch. 6.4 - A tank is full of water. Find the work required to...Ch. 6.4 - Suppose that for the tank in Exercise 23 the pump...Ch. 6.4 - Solve Exercise 24 if the tank is half full of oil...Ch. 6.4 - When gas expands in a cylinder with radius r, the...Ch. 6.4 - In a steam engine the pressure P and volume V of...Ch. 6.4 - Prob. 31ECh. 6.4 - Prob. 32ECh. 6.4 - (a) Newtons Law of Gravitation states that two...Ch. 6.4 - The Great Pyramid of King Khufu was built of...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - Find the average value of the function on the...Ch. 6.5 - (a) Find the average value of f on the given...Ch. 6.5 - (a) Find the average value of f on the given...Ch. 6.5 - (a) Find the average value of f on the given...Ch. 6.5 - Prob. 12ECh. 6.5 - If f is continuous and 13f(x)dx=8, show that f...Ch. 6.5 - Find the numbers b such that the average value of...Ch. 6.5 - Find the average value of f on [0, 8].Ch. 6.5 - The velocity graph of an accelerating car is...Ch. 6.5 - In a certain city the temperature (in F) t hours...Ch. 6.5 - The velocity v of blood that flows in a blood...Ch. 6.5 - The linear density in a rod 8 m long is...Ch. 6.5 - (a) A cup of coffee has temperature 95C and takes...Ch. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Use the result of Exercise 5.5.83 to compute the...Ch. 6.5 - Use the diagram to show that if f is concave...Ch. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6 - (a) Draw two typical curves y = f(x) and y = g(x),...Ch. 6 - Suppose that Sue runs faster than Kathy throughout...Ch. 6 - Prob. 3RCCCh. 6 - Prob. 4RCCCh. 6 - Suppose that you push a book across a 6-meter-long...Ch. 6 - Prob. 6RCCCh. 6 - Find the area of the region bounded by the given...Ch. 6 - Find the area of the region bounded by the given...Ch. 6 - Prob. 3RECh. 6 - Find the area of the region bounded by the given...Ch. 6 - Find the area of the region bounded by the given...Ch. 6 - Find the area of the region bounded by the given...Ch. 6 - Prob. 7RECh. 6 - Find the volume of the solid obtained by rotating...Ch. 6 - Find the volume of the solid obtained by rotating...Ch. 6 - Find the volume of the solid obtained by rotating...Ch. 6 - Prob. 11RECh. 6 - Set up, but do not evaluate, an integral for the...Ch. 6 - Prob. 13RECh. 6 - Set up, but do not evaluate, an integral for the...Ch. 6 - Find the volumes of the solids obtained by...Ch. 6 - Let be the region in the first quadrant bounded...Ch. 6 - Prob. 17RECh. 6 - Let be the region bounded by the curves y = 1 x2...Ch. 6 - Prob. 19RECh. 6 - Each integral represents the volume of a solid....Ch. 6 - Each integral represents the volume of a solid....Ch. 6 - Each integral represents the volume of a solid....Ch. 6 - The base of a solid is a circular disk with radius...Ch. 6 - The base of a solid is the region bounded by the...Ch. 6 - Prob. 25RECh. 6 - Prob. 26RECh. 6 - Prob. 27RECh. 6 - A 1600-lb elevator is suspended by a 200-ft cable...Ch. 6 - A tank full of water has the shape of a paraboloid...Ch. 6 - A steel tank has the shape of a circular cylinder...Ch. 6 - Prob. 31RECh. 6 - Prob. 32RECh. 6 - Prob. 33RECh. 6 - Prob. 34RECh. 6 - Prob. 1PCh. 6 - There is a line through the origin that divides...Ch. 6 - The figure shows a horizontal line y = c...Ch. 6 - A cylindrical glass of radius r and height L is...Ch. 6 - (a) Show that the volume of a segment of height h...Ch. 6 - Archimedes Principle states that the buoyant force...Ch. 6 - Prob. 7PCh. 6 - Prob. 8PCh. 6 - The figure shows a curve C with the property that,...Ch. 6 - A paper drinking cup filled with water has the...Ch. 6 - A clepsydra, or water clock, is a glass container...Ch. 6 - A cylindrical container of radius r and height L...Ch. 6 - Prob. 13PCh. 6 - If the tangent at a point P on the curve y = x3...
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