Calculus, Early Transcendentals
9th Edition
ISBN: 9781337613927
Author: Stewart
Publisher: CENGAGE L
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Chapter 8.2, Problem 28E
Find the exact area of the surface obtained by rotating the given curve about the
28.
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4. Suppose that A is made up of 5 column vectors in R³, and suppose that the
rank(A)=3.
a. How many solutions are there to Ax=0? Justify.
b. What is a geometric description for the nullspace(A)? Justify.
c. Do the column vectors of A span R³? Justify.
d. Is A invertible? Justify.
3. Suppose that A is 5 x 5 and rank(A)=4. Use this information to answer the following.
a. Give a geometric description of nullspace(A). Justify.
b. Is A invertible? Justify.
c. Give a geometric description of the span of the column vectors of A. What
space are the column vectors of A in? Justify.
d. What is determinant of A? Justify.
2. Consider the matrix:
A
||
1
1
-3
14
2 1 01
4 1
2
2
-26
1 -3
1 5]
a) What is rank(A)?
b) Is A invertible? Justify.
c) Find the nullspace(A). Justify.
d) Is the trivial solution the only solution to Ax=0? Justify.
e) What is the span of the column vectors of A? Justify.
Chapter 8 Solutions
Calculus, Early Transcendentals
Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Set up, but do not evaluate, an integral for the...Ch. 8.1 - Find the exact length of the curve. 9....Ch. 8.1 - Find the exact length of the curve. 10....
Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 12....Ch. 8.1 - Find the exact length of the curve. 15....Ch. 8.1 - Find the exact length of the curve. 14. y = ln(cos...Ch. 8.1 - Find the exact length of the curve. 15. y = ln(sec...Ch. 8.1 - Prob. 18ECh. 8.1 - Find the exact length of the curve. 19....Ch. 8.1 - Find the exact length of the curve. 16....Ch. 8.1 - Find the exact length of the curve. 17....Ch. 8.1 - Find the exact length of the curve. 18....Ch. 8.1 - Find the exact length of the curve. 19....Ch. 8.1 - Find the exact length of the curve. 20. y = 1 ex,...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Prob. 37ECh. 8.1 - Use either a computer or a table of integrals to...Ch. 8.1 - Prob. 39ECh. 8.1 - (a) Sketch the curve y3 = x2. (b) Use Formulas 3...Ch. 8.1 - Find the arc length function for the curve y =...Ch. 8.1 - (a) Find the arc length function for the curve y =...Ch. 8.1 - Find the arc length function for the curve...Ch. 8.1 - The arc length function for a curve y = f(x),...Ch. 8.1 - A steady wind blows a kite due west. The kites...Ch. 8.1 - Prob. 49ECh. 8.1 - Prob. 53ECh. 8.1 - The curves shown are all examples of graphs of...Ch. 8.2 - The given curve is rotated about the x -axis. Set...Ch. 8.2 - The given curve is rotated about the x -axis. Set...Ch. 8.2 - The given curve is rotated about the x -axis. Set...Ch. 8.2 - The given curve is rotated about the x -axis. Set...Ch. 8.2 - The given curve is rotated about the y -axis. Set...Ch. 8.2 - The given curve is rotated about the y -axis. Set...Ch. 8.2 - The given curve is rotated about the y -axis. Set...Ch. 8.2 - Prob. 8ECh. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - If the infinite curve y = ex, x 0, is rotated...Ch. 8.2 - (a) The ellipse x2a2+y2b2=1ab is rotated about the...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - Prob. 44ECh. 8.2 - Show that if we rotate the curve y = ex/2 + e x/2...Ch. 8.2 - Formula 4 is valid only when f(x) 0. Show that...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - A tank is 8 m long, 4 m wide, 2 m high, and...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - A cube with 20-cm-long sides is sitting on the...Ch. 8.3 - Prob. 16ECh. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Suppose that a plate is immersed vertically in a...Ch. 8.3 - A metal plate was found submerged vertically in...Ch. 8.3 - (a) Use the formula of Exercise 18 to show that...Ch. 8.3 - Point-masses mi arc located on the x-axis as...Ch. 8.3 - Point-masses mi arc located on the x-axis as...Ch. 8.3 - Prob. 23ECh. 8.3 - Prob. 24ECh. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Use Simpsons Rule to estimate the centroid of the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Prob. 38ECh. 8.3 - Prove that the centroid of any triangle is located...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - A rectangle with sides a and b is divided into...Ch. 8.3 - Prob. 43ECh. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Let be the region that lies between the curves...Ch. 8.3 - Prove Formulas 9.Ch. 8.4 - The marginal cost function C(x) was defined to be...Ch. 8.4 - A company estimates that the marginal revenue (in...Ch. 8.4 - Prob. 3ECh. 8.4 - The demand function for a particular vacation...Ch. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 15ECh. 8.4 - Prob. 16ECh. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Use Poiseuilles Law to calculate the rate of flow...Ch. 8.4 - Prob. 22ECh. 8.4 - After a 5.5-mg injection of dye, the readings of...Ch. 8.4 - The graph of the concentration function c(t) is...Ch. 8.5 - Let f(x) be the probability density function for...Ch. 8.5 - Let f(t) be the probability density function for...Ch. 8.5 - Let .f(x) = 30x2(1 x)2 for 0 x 1 and f(x) = 0...Ch. 8.5 - The density function f(x)=e3x(1+e3x)2 is an...Ch. 8.5 - Let f(x) = c/(1 + x2). (a) For what value of c is...Ch. 8.5 - Let f(x) = k(3x x2) if 0 x 3 and f(x) = 0 if x ...Ch. 8.5 - A spinner from a board game randomly indicates a...Ch. 8.5 - (a) Explain why the function whose graph is shown...Ch. 8.5 - Show that the median waiting time for a phone call...Ch. 8.5 - (a) A type of light bulb is labeled as having an...Ch. 8.5 - An online retailer has determined that the average...Ch. 8.5 - Prob. 12ECh. 8.5 - REM sleep is the phase of sleep when most active...Ch. 8.5 - According to the National Health Survey, the...Ch. 8.5 - The Garbage Project at the University of Arizona...Ch. 8.5 - Boxes are labeled as containing 500 g of cereal....Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - Prob. 20ECh. 8.5 - The hydrogen atom is composed of one proton in the...Ch. 8 - (a) How is the length of a curve defined? (b)...Ch. 8 - Prob. 2CCCh. 8 - Describe how we can find the hydrostatic force...Ch. 8 - Prob. 4CCCh. 8 - Prob. 5CCCh. 8 - Prob. 6CCCh. 8 - Prob. 7CCCh. 8 - Prob. 8CCCh. 8 - Prob. 9CCCh. 8 - Prob. 10CCCh. 8 - Prob. 1TFQCh. 8 - Prob. 2TFQCh. 8 - Prob. 4TFQCh. 8 - Prob. 5TFQCh. 8 - Prob. 6TFQCh. 8 - Prob. 8TFQCh. 8 - Find the length of the curve. y = 4(x 1)3/2, 1 x...Ch. 8 - Find the length of the curve. y=2ln(sin12x),/3xCh. 8 - Find the length of the curve. 12x = 4y3 + 3y1, 1 ...Ch. 8 - (a) Find the length of the curve y=x416+12x21x2...Ch. 8 - Prob. 5ECh. 8 - (a) The curve y = x2, 0 x 1, is rotated about...Ch. 8 - Use Simpson's Rule with n = 10 to estimate the...Ch. 8 - Find the length of the curve y=1xt1dt1x16Ch. 8 - Prob. 10ECh. 8 - A gate in an irrigation canal is constructed in...Ch. 8 - A trough is filled with water and its vertical...Ch. 8 - Find the centroid of the region shown.Ch. 8 - Find the centroid of the region shown.Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Prob. 17ECh. 8 - Use the Theorem of Pappus and the fact that the...Ch. 8 - The demand function for a commodity is given by p...Ch. 8 - After a 6-mg injection of dye into a heart, the...Ch. 8 - Prob. 21ECh. 8 - Prob. 22ECh. 8 - Prob. 23ECh. 8 - Prob. 1PPCh. 8 - Find the centroid of the region enclosed by the...Ch. 8 - If a sphere of radius r is sliced by a plane whose...Ch. 8 - Prob. 4PPCh. 8 - Prob. 5PPCh. 8 - The figure shows a semicircle with radius 1,...Ch. 8 - Prob. 7PPCh. 8 - Consider a flat metal plate to be placed...Ch. 8 - A uniform disk with radius 1 m is to be cut by a...Ch. 8 - A triangle with area 30 cm2 is cut from a corner...Ch. 8 - Prob. 11PPCh. 8 - If the needle in Problem 11 has length h L, its...Ch. 8 - Find the centroid of the region enclosed by the...
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