Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 8.1, Problem 44E
(a) The figure shows a telephone wire hanging between two poles at x = – b and x = b. It takes the shape of a catenary with equation y = c + a cosh(x/a). Find the length of the wire.
(b) Suppose two telephone poles are 50ft apart and the length of the wire between the poles is 51 ft. If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?
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Please solve 14 and 15
1. Consider the following system of equations:
x13x2 + 4x3 - 5x4 = 7
-2x13x2 + x3 - 6x4 = 7
x16x213x3 - 21x4 = 28
a) Solve the system. Write your solution in parametric and vector form.
b) What is a geometric description of the solution.
7
c) Is v =
7 in the span of the set S=
[28.
1
HE
3
-5
3
·6
? If it is, write v
6
as a linear combination of the vectors in S. Justify.
d) How many solutions are there to the associated homogeneous system for
the system above? Justify.
e) Let A be the coefficient matrix from the system above. Find the set of all
solutions to Ax = 0.
f) Is there a solution to Ax=b for all b in R³? Justify.
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.
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 an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...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. 13....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 - 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 - 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 - 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. 29ECh. 8.1 - Prob. 30ECh. 8.1 - Sketch the curve with equation x2/3 + y2/3 = 1 and...Ch. 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 - For the function f(x)=14ex+ex, prove that the arc...Ch. 8.1 - A steady wind blows a kite due west. The kites...Ch. 8.1 - A hawk flying at 15m/s at an altitude of 180 m...Ch. 8.1 - The Gateway Arch in St. Louis (see the photo on...Ch. 8.1 - A manufacturer of corrugated metal roofing wants...Ch. 8.1 - (a) The figure shows a telephone wire hanging...Ch. 8.1 - Prob. 45ECh. 8.1 - The curves with equations xn + yn = 1, n = 4, 6,...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...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 - 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 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Prob. 21ECh. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - If the region = {(x, y) | x 1, 0 y 1/x} is...Ch. 8.2 - If the infinite curve y = ex, x 0, is rotated...Ch. 8.2 - Prob. 29ECh. 8.2 - A group of engineers is building a parabolic...Ch. 8.2 - (a) The ellipse x2a2+y2b2=1ab is rotated about the...Ch. 8.2 - Find the surface area of the torus in Exercise...Ch. 8.2 - If the curve y = f(x), a x b, is rotated about...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - (a) Show that the surface area of a zone of a...Ch. 8.2 - Show that if we rotate the curve y = ex/2 + e x/2...Ch. 8.2 - Prob. 38ECh. 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 plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 8.3 - A trough is filled with a liquid of density 840...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 - The masses mi, are located at the points Pi. Find...Ch. 8.3 - The masses mi, are located at the points Pi. Find...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...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 - 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 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - The centroid of a curve can be found by a process...Ch. 8.3 - The Second Theorem of Pappus is in the same spirit...Ch. 8.3 - Use the Second Theorem of Pappus described in...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 - A demand curve is given by p = 450/(x + 8). Find...Ch. 8.4 - The supply function ps(x) for a commodity gives...Ch. 8.4 - If a supply curve is modeled by the equation p =...Ch. 8.4 - In a purely competitive market, the price of a...Ch. 8.4 - The sum of consumer surplus and producer surplus...Ch. 8.4 - A camera company estimates that the demand...Ch. 8.4 - A company modeled the demand curve for its product...Ch. 8.4 - A movie theater has been charging 10.00 per...Ch. 8.4 - Prob. 13ECh. 8.4 - Prob. 14ECh. 8.4 - Prob. 15ECh. 8.4 - Prob. 16ECh. 8.4 - Prob. 17ECh. 8.4 - Prob. 18ECh. 8.4 - Use Poiseuilles Law to calculate the rate of flow...Ch. 8.4 - Prob. 20ECh. 8.4 - The dye dilution method is used to measure cardiac...Ch. 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. 2RCCCh. 8 - Describe how we can find the hydrostatic force...Ch. 8 - Prob. 4RCCCh. 8 - Prob. 5RCCCh. 8 - Prob. 6RCCCh. 8 - Prob. 7RCCCh. 8 - Prob. 8RCCCh. 8 - Prob. 9RCCCh. 8 - Prob. 10RCCCh. 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. 5RECh. 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 - Prob. 8RECh. 8 - Find the length of the curve y=1xt1dt1x16Ch. 8 - Prob. 10RECh. 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. 17RECh. 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. 21RECh. 8 - Prob. 22RECh. 8 - Prob. 23RECh. 8 - Prob. 1PCh. 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. 4PCh. 8 - Prob. 5PCh. 8 - The figure shows a semicircle with radius 1,...Ch. 8 - Prob. 7PCh. 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. 11PCh. 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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- 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.arrow_forward2. 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.arrow_forwardE 5. Suppose that S={v € R²: v = [2x² - 3]}. Is S a subspace of R²? Prove or disprovearrow_forward
- 6. Suppose that V1, V2 ER", show that span{v1, v2} is a subspace of Rn.arrow_forwardRa X 2) slots per pole per phase 3/31 180 Ko Sin (1) Kdl 1 sin (4) sin(3) Sin (30) اذا مرید شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed, 120 Fo lasa! G s.1000-950 20:05 1000 Capper losses: 5kw Rotor input lookw 0.05 ined sove in peaper I need a detailed solution on paper please 6) 1 ۳/۱ وه اذا ميريد شرح الكتب فقط look DC 7) rotov Find the general solution of the following equations: +4y=tan2x 3 7357 Find the general solution of the following equations: - Qll y + y (³) = 0. 101arrow_forwardB: 18060 msl Kd Ka, Sin (n) I sin () sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed, 120 x 50 G 5005 1000 s = 1000-950 Copper bosses 5kW /0001 Rotor input 5 : loo kw 0.05 6) 1 اذا ميريد شرح الكتب فقط ١٥٠ 7) rotov DC ined sove in Deaper I need a detailed solution on paper please dy x+2y-4 = dx 2x-y-3 Find the general solution of the following equations: 02//yl-4y+13y=esinarrow_forward
- 1) R₂ = X2 2) slots per pole per phase = 3/31 B msl kd 180 60 Kal Sin (1) I sin () sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed, 120 x 50 G 5005 1000 s = 1000-950 Copper bosses 5kW Rotor input: 5 0.05 loo kw 6) 1 /0001 اذا ميريد شرح الكتب فقط look 7) rotov DC ined sove in peaper I need a detailed solution on paper please Q1// Find the solution of: 'y' = x² +376 x4+316 xyo Q2 Find the solution of the initial-valued problems: ex-y y' +exarrow_forwardR₂ = X2 2) slots per pole per phase = 3/31 B-18060 msl kd Kasi Sin (1) I sin (6) sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed s = 1000-950 1000 Copper losses: 5kw Rotor input 5 0.05 6) 1 120 x 50 G loo kw ined sove in peaper I need a detailed solution on paper please Q3// x²y// +xy/ + (x² - ½) y = x³/². اذا ميريد شرح الكتب فقط look 7) rotor DC Q4// x²y// - (2x+x²)y/ + (2 + x)y = x³. dy 2x+2y+4 = dx 2x-y-3arrow_forward۳/۱ R2X2 2) slots per pole per phase = 3/31 B, 18060 msl Kas Sin() 1sin() sin(30) Sin (30) kd اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speeds S = 1000-950 1000 Copper bosses 5kw 120*50 loca G Rotor input 5 loo kw 6) 1 0.05 اذا ميريد شرح الكتب فقط lookw 7) rotor DC ined sove in peaper I need a detailed solution on paper please 064 Q1// Find the solution of QI/Find the solution of Inxy= 7357 x+2y³ y' = xy3arrow_forward
- R₂ = X2 2) slots per pole per phase 3/31 msl 180 60 Kd Ka Sin (1) Isin (6) sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed, 120*50 1000 6 S = 1000-950 1000 Copper bosses: 5kw Rotor input 5 0.05 : loo kw 6) 1 اذا ميريد شرح الكتب فقط 100 7) rotor DC ined sove in peaper I need a detailed solution on paper please Find the general solution of the following equations: Q2lyl-4y+13y=esinx. Find the general solution of the following equations: " Qly (49) - 16y= 0. 151arrow_forward۳/۱ R₂ = X2 2) slots per pole per phase = 3/31 B-18060 msl kd Kasi Sin (1) I sin (6) sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speed s = 1000-950 1000 Copper losses: 5kw Rotor input 5 0.05 6) 1 120 x 50 G loo kw اذا میرید شرح الكتب فقط look 7) rotor DC ined sove in peaper I need a detailed solution on paper dy please 04 12=-cosx.y + 2cosx with y(x) = 1 か 'Oy + xlny + xe")dx + (xsiny + xlnx +*dy=0. 01arrow_forward٣/١ R2X2 2) slots per pole per phase = 3/31 B, 18060 msl kd Kas Sin (1) 1sin() sin(30) Sin (30) اذا ميريد شرح الكتب بس 0 بالفراغ 3) Cos (30) 0.866 4) Rotating 5) Synchronous speeds S = 1000-950 1000 Copper bosses 5kw 120*50 loca G Rotor input 5 loo kw 0.05 6) 1 اذا ميريد شرح الكتب فقط lookw 7) rotor ined sove in peaper I need a detailed solution on paper please DC 口 04 on its wheels as shown in figure. The the door is 8 m below the free surface o is located at the center of the d no water leaks an accident and lands at the bottom of the lake 12m high and I m wide, and the top edge of water Determine the hydrostatic force on the discuss if the driver can open the door, if ong person can lift 100 kg, the passenger The door can be approximated as a vertical rec | 279|-|(23+2+12+20=2) AA Find the general solution of the following equations: 11 - 1/4+xy/-(1-x²³)= 0. 2arrow_forward
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