Calculus: Early Transcendentals
12th Edition
ISBN: 9781119778189
Author: Anton, Howard, Bivens, Irl C., Davis, Stephen
Publisher: WILEY
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Chapter 12.2, Problem 53ES
Use Formula (7) to derive the
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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.
Chapter 12 Solutions
Calculus: Early Transcendentals
Ch. 12.1 - Prob. 1QCECh. 12.1 - Describe the graph of rt=1+2t,1+3t.Ch. 12.1 - Prob. 3QCECh. 12.1 - Prob. 4QCECh. 12.1 - Prob. 1ESCh. 12.1 - Find the domain of r(t) and the value of rt0. 2....Ch. 12.1 - Find the domain of rt and the value of rt0....Ch. 12.1 - Find the domain of rt and the value of rt0....Ch. 12.1 - Prob. 5ESCh. 12.1 - Prob. 6ES
Ch. 12.1 - Prob. 7ESCh. 12.1 - Prob. 8ESCh. 12.1 - Describe the graph of the equation. r=32ti+5tjCh. 12.1 - Describe the graph of the equation....Ch. 12.1 - Describe the graph of the equation. r=2ti3j+1+3tkCh. 12.1 - Prob. 12ESCh. 12.1 - Describe the graph of the equation....Ch. 12.1 - Describe the graph of the equation. r=3i+1t2j+tkCh. 12.1 - (a) Find the slope of the line in 2-space that is...Ch. 12.1 - (a) Find the y-intercept of the line in 2-space...Ch. 12.1 - Prob. 17ESCh. 12.1 - Prob. 18ESCh. 12.1 - Write a vector equation for the line segment from...Ch. 12.1 - Write a vector equation for the line segment from...Ch. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Prob. 22ESCh. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Prob. 25ESCh. 12.1 - Prob. 26ESCh. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Sketch the graph of rt and show the direction of...Ch. 12.1 - Prob. 31ESCh. 12.1 - Determine whether the statement is true or false....Ch. 12.1 - Sketch the curve of intersection of the surfaces,...Ch. 12.1 - Sketch the curve of intersection of the surfaces,...Ch. 12.1 - Sketch the curve of intersection of the surfaces,...Ch. 12.1 - Prob. 38ESCh. 12.1 - Show that the graph of r=tsinti+tcostj+t2k lies on...Ch. 12.1 - Prob. 41ESCh. 12.1 - How many revolutions will the circular helix...Ch. 12.1 - Show that the curve r=tcosti+tsintj+tk,t0, lies on...Ch. 12.1 - Describe the curve r=acosti+bsintj+ctk, where...Ch. 12.1 - In each part, match the vector equation with one...Ch. 12.1 - (a) Find parametric equations for the curve of...Ch. 12.1 - (a) Sketch the graph of rt=2t,21+t2 (b) Prove that...Ch. 12.1 - Prob. 51ESCh. 12.1 - Suppose that r1tandr2t are vector-valued functions...Ch. 12.2 - alimt3t2i+2tj=blimt/4cost,sint=Ch. 12.2 - Find r t. art=4+5ti+tt2jbrt=1t,tant,e2tCh. 12.2 - Suppose that r10=3,2,1,r20=1,2,3,r 10=0,0,0,andr...Ch. 12.2 - a012t,t2,sintdt=bti3t2j+etkdt=Ch. 12.2 - Find the limit. limt+t2+13t2+2,1tCh. 12.2 - Find the limit. limt2ti3j+t2kCh. 12.2 - Determine whether rt is continuous at t=0. Explain...Ch. 12.2 - Determine whether rt is continuous at t=0. Explain...Ch. 12.2 - Sketch the circle rt=costi+sintj, draw the vector...Ch. 12.2 - Find r t. rt=4icostjCh. 12.2 - Find r t. rt=tan1ti+tcostjtkCh. 12.2 - Prob. 11ESCh. 12.2 - Prob. 12ESCh. 12.2 - Prob. 13ESCh. 12.2 - Find the vector rt0; then sketch the graph of rt...Ch. 12.2 - Find the vector rt0; then sketch the graph of rt...Ch. 12.2 - Prob. 16ESCh. 12.2 - Find parametric equations of the line tangent to...Ch. 12.2 - Find parametric equations of the line tangent to...Ch. 12.2 - Prob. 23ESCh. 12.2 - Find a vector equation of the line tangent to the...Ch. 12.2 - Prob. 25ESCh. 12.2 - Prob. 26ESCh. 12.2 - Letrt=costi+sintj+k.findalimt0rtr tblimt0rtr...Ch. 12.2 - Prob. 28ESCh. 12.2 - Calculate ddtr1tr2tandddtr1tr2t first by...Ch. 12.2 - Calculate ddtr1tr2tandddtr1tr2t first by...Ch. 12.2 - Evaluate the indefinite integral. 3i+4tjdtCh. 12.2 - Evaluate the indefinite integral. tet,lntdtCh. 12.2 - Evaluate the indefinite integral. et,et,3t2dtCh. 12.2 - Evaluate the definite integral. 0/2cos2t,sin2tdtCh. 12.2 - Evaluate the definite integral. 02ti+t2jdtCh. 12.2 - Evaluate the definite integral. 19t1/2i+t1/2jdtCh. 12.2 - Evaluate the definite integral. 01e2ti+etj+tkdtCh. 12.2 - Prob. 41ESCh. 12.2 - Prob. 43ESCh. 12.2 - Solve the vector initial-value problem for yt by...Ch. 12.2 - Solve the vector initial-value problem for yt by...Ch. 12.2 - Solve the vector initial-value problem for yt by...Ch. 12.2 - (a) Find the points where the curve r=ti+t2j3tk...Ch. 12.2 - Show that the graphs of r1tandr2t intersect at the...Ch. 12.2 - Show that the graphs of r1tandr2t intersect at the...Ch. 12.2 - Use Formula (7) to derive the differentiation...Ch. 12.2 - Let u=ut,v=vt,andw=wt be differentiable...Ch. 12.2 - Let u1,u2,u3,1,2,3,w1,w2,andw3, be differentiable...Ch. 12.2 - Prove Theorem 12.2.6 for 2-space.Ch. 12.2 - Derive Formulas (6) and (7) for 3-space.Ch. 12.2 - Prove Theorem 12.2.9 for 2-space.Ch. 12.2 - Prob. 59ESCh. 12.2 - Prob. 60ESCh. 12.3 - If rt is a smooth vector-valued function, then the...Ch. 12.3 - If r(s) is a smooth vector-valued function...Ch. 12.3 - If rt is a smooth vector-valued function, then the...Ch. 12.3 - Suppose that rt is a smooth vector-valued function...Ch. 12.3 - Determine whether rt is a smooth function of the...Ch. 12.3 - Determine whether rt is a smooth function of the...Ch. 12.3 - Determine whether rt is a smooth function of the...Ch. 12.3 - Find the arc length of the parametric curve....Ch. 12.3 - Find the arc length of the parametric curve....Ch. 12.3 - Find the arc length of the graph of rt....Ch. 12.3 - Find the arc length of the graph of rt....Ch. 12.3 - Calculate dr/d by the chain rule, and then check...Ch. 12.3 - Calculate dr/d by the chain rule, and then check...Ch. 12.3 - Calculate dr/d by the chain rule, and then check...Ch. 12.3 - (a) Find the arc length parametrization of the...Ch. 12.3 - Find arc length parametrizations of the lines in...Ch. 12.3 - Prob. 23ESCh. 12.3 - (a) Find the arc length parametrization of the...Ch. 12.3 - Find an arc length parametrization of the curve...Ch. 12.3 - Find an arc length parametrization of the curve...Ch. 12.3 - Find an arc length parametrization of the curve...Ch. 12.3 - Find an arc length parametrization of the curve...Ch. 12.3 - Find an arc length parametrization of the curve...Ch. 12.3 - Show that the arc length of the circular helix...Ch. 12.3 - Use the result in Exercise 31 to show the circular...Ch. 12.3 - Find an arc length parametrization of the cycloid...Ch. 12.3 - Show that in cylindrical coordinates a curve given...Ch. 12.3 - In each part, use the formula in Exercise 34 to...Ch. 12.3 - Show That in spherical coordinates a curve given...Ch. 12.3 - In each part, use the formula in Exercise 36 to...Ch. 12.3 - h (a) Sketch the graph of rt=ti+t2j. Show that rt...Ch. 12.3 - Find a change of parameter t=g for the semicircle...Ch. 12.3 - What change of parameter t=g would you make if you...Ch. 12.3 - As illustrated in the accompanying figure, copper...Ch. 12.3 - Let rt=lnti+2tj+t2k.Findartbdsdtc13rtdt.Ch. 12.3 - Let rt=t2i+t3j (seeFigure12.3.1) . Let t be the...Ch. 12.3 - Prove: If rt is a smoothly parametrized function,...Ch. 12.3 - Prove the vector form of the chain rule for...Ch. 12.3 - Prob. 47ESCh. 12.4 - Prob. 1QCECh. 12.4 - If C is the graph of a smooth vector-valued...Ch. 12.4 - If C is the graph of a smooth vector-valued...Ch. 12.4 - Suppose that C is the graph of a smooth...Ch. 12.4 - In each part, sketch the unit tangent and normal...Ch. 12.4 - In the marginal note associated with Example 8 of...Ch. 12.4 - Prob. 4ESCh. 12.4 - Find TtandNt at the given point. rt=t21i+tj;t=1Ch. 12.4 - Find TtandNt at the given point....Ch. 12.4 - Find TtandNt at the given point....Ch. 12.4 - Prob. 10ESCh. 12.4 - Find TtandNt at the given point....Ch. 12.4 - Find TtandNt at the given point....Ch. 12.4 - Use the result in Exercise 3 to find parametric...Ch. 12.4 - Use the result in Exercise 3 to find parametric...Ch. 12.4 - Use the formula Bt=TtNt to find Bt, and then check...Ch. 12.4 - Use the formula Bt=TtNt to find Bt, and then check...Ch. 12.4 - Use the formula Bt=TtNt to find Bt, and then check...Ch. 12.4 - Prob. 18ESCh. 12.4 - Find Tt,Nt,andBt for the given value of t. Then...Ch. 12.4 - Find Tt,Nt,andBt for the given value of t. Then...Ch. 12.4 - Prob. 21ESCh. 12.4 - Prob. 22ESCh. 12.4 - Prob. 24ESCh. 12.4 - Prob. 25ESCh. 12.4 - Discuss some of the advantages of parametrizing a...Ch. 12.5 - If C is a smooth curve parametrized by arc length,...Ch. 12.5 - Let rt be a smooth vector-valued function with...Ch. 12.5 - Suppose that C is the graph of a smooth...Ch. 12.5 - Suppose that C is a smooth curve and that x2+y2=4...Ch. 12.5 - Use the osculating circle shown in the figure to...Ch. 12.5 - For a plane curve y=fx the curvature at x,fx is...Ch. 12.5 - For a plane curve y=fx the curvature at x,fx is...Ch. 12.5 - Use Formula (3) to find t. rt=t2i+t3jCh. 12.5 - Use Formula (3) to find t. rt=e3ti+etjCh. 12.5 - Use Formula (3) to find t. rt=4costi+4sintj+tkCh. 12.5 - Use Formula (3) to find t. x=cosht,y=sinht,z=tCh. 12.5 - Find the curvature and the radius of curvature at...Ch. 12.5 - Find the curvature and the radius of curvature at...Ch. 12.5 - Confirm that s is an arc length parameter by...Ch. 12.5 - Prob. 19ESCh. 12.5 - Determine whether the statement is true or false....Ch. 12.5 - Prob. 22ESCh. 12.5 - (a) Use Formula (3) to show that in 2-space the...Ch. 12.5 - Use part (b) of Exercise 23 to show that the...Ch. 12.5 - Use the result in Exercise 23(b) to find the...Ch. 12.5 - Use the result in Exercise 23(a) to find the...Ch. 12.5 - Use the result in Exercise 23(a) to find the...Ch. 12.5 - Use the result in Exercise 23(a) to find the...Ch. 12.5 - In each part, use the formulas in Exercise 23 to...Ch. 12.5 - Prob. 34ESCh. 12.5 - Generate the graph of y=fx using a graphing...Ch. 12.5 - Generate the graph of y=fx using a graphing...Ch. 12.5 - (a) Use a CAS to graph the parametric curve...Ch. 12.5 - Use the formula in Exercise 23 (a) to show that...Ch. 12.5 - Use the result in Exercise 39 to show that a...Ch. 12.5 - Find the radius of curvature of the parabola...Ch. 12.5 - At what point(s) does 4x2+9y2=36 have a minimum...Ch. 12.5 - Find the maximum and minimum values of the radius...Ch. 12.5 - Use the formula in Exercise 39 to show that the...Ch. 12.5 - Use the formula in Exercise 39 and a CAS to show...Ch. 12.5 - Prob. 51ESCh. 12.5 - The evolute of a smooth parametric curve C in...Ch. 12.5 - These exercises are concerned with the problem of...Ch. 12.5 - These exercises are concerned with the problem of...Ch. 12.5 - These exercises are concerned with the problem of...Ch. 12.5 - These exercises are concerned with the problem of...Ch. 12.5 - These exercises are concerned with the problem of...Ch. 12.5 - Assume that s is an arc length parameter for a...Ch. 12.5 - Assume that s is an arc length parameter for a...Ch. 12.5 - Assume that s is an arc length parameter for a...Ch. 12.5 - Prob. 61ESCh. 12.5 - (a) Use the chain rule and the first two...Ch. 12.5 - Use the formula in Exercise 62(d) to find the...Ch. 12.5 - Use the formula in Exercise 62(d) to find the...Ch. 12.5 - The accompanying figure is the graph of the radius...Ch. 12.6 - If r(t) is the position function of a particle,...Ch. 12.6 - If r(t) is the position function of a particle,...Ch. 12.6 - The tangential scalar component of acceleration is...Ch. 12.6 - The projectile motion model r(t)=12gt2+s0j+tv0...Ch. 12.6 - Prob. 1ESCh. 12.6 - Prob. 2ESCh. 12.6 - Prob. 3ESCh. 12.6 - Prob. 4ESCh. 12.6 - Find the velocity, speed, and acceleration at the...Ch. 12.6 - Find the velocity, speed, and acceleration at the...Ch. 12.6 - Find the velocity, speed, and acceleration at the...Ch. 12.6 - Prob. 8ESCh. 12.6 - As illustrated in the accompanying figure, suppose...Ch. 12.6 - Prob. 10ESCh. 12.6 - What can you say about the trajectory of a...Ch. 12.6 - Prob. 12ESCh. 12.6 - Suppose that the position vector of a particle...Ch. 12.6 - Prob. 14ESCh. 12.6 - Prob. 17ESCh. 12.6 - Prob. 18ESCh. 12.6 - Prob. 19ESCh. 12.6 - Prob. 20ESCh. 12.6 - Find to the nearest degree, the angle between v...Ch. 12.6 - Prob. 22ESCh. 12.6 - Prob. 23ESCh. 12.6 - Find the displacement and the distance travelled...Ch. 12.6 - Prob. 26ESCh. 12.6 - Find the displacement and the distance travelled...Ch. 12.6 - Prob. 28ESCh. 12.6 - The position vectors r1andr2 of two particles are...Ch. 12.6 - The position vectors r1andr2 of two particles are...Ch. 12.6 - Prob. 31ESCh. 12.6 - Prob. 32ESCh. 12.6 - Prob. 33ESCh. 12.6 - The position function of a particle is given. Use...Ch. 12.6 - Prob. 35ESCh. 12.6 - Prob. 36ESCh. 12.6 - Prob. 37ESCh. 12.6 - In these exercises v and a are given at a certain...Ch. 12.6 - Prob. 39ESCh. 12.6 - The speed v of a particle at an arbitrary time t...Ch. 12.6 - The nuclear accelerator at the Enrico Fermi...Ch. 12.6 - Prob. 42ESCh. 12.6 - Prob. 43ESCh. 12.6 - Prob. 44ESCh. 12.6 - Use the given information to find the normal...Ch. 12.6 - Prob. 46ESCh. 12.6 - Determine whether the statement is true or false....Ch. 12.6 - Determine whether the statement is true or false....Ch. 12.6 - Determine whether the statement is true or false....Ch. 12.6 - Determine whether the statement is true or false....Ch. 12.6 - Derive Formula (18) from Formula (14).Ch. 12.6 - If an automobile of mass m rounds a curve, then...Ch. 12.6 - A Shell is fired from ground level with a muzzle...Ch. 12.6 - A rock is thrown downward from the top of a...Ch. 12.6 - Solve Exercise 55 assuming that the rock is thrown...Ch. 12.6 - Prob. 57ESCh. 12.6 - A shell, fired from ground level at an elevation...Ch. 12.6 - Find two elevation angles that will enable a...Ch. 12.6 - A ball rolls off a table 4 ft high while moving at...Ch. 12.6 - As illustrated in the accompanying figure, a fire...Ch. 12.6 - What is the minimum initial velocity that will...Ch. 12.6 - As shown in the accompanying figure on the next...Ch. 12.6 - As illustrated in the accompanying figure, a train...Ch. 12.6 - A shell is fired from ground level at an elevation...Ch. 12.6 - A shell is fired from ground level with an...Ch. 12.6 - At time t=0 a baseball that is 5 ft above the...Ch. 12.6 - Repeat Exercise 67, assuming that the ball leaves...Ch. 12.6 - At time t=0 a skier leaves the end of a ski jump...Ch. 12.6 - At time t=0 a projectile is fired from a height h...Ch. 12.6 - Prob. 71ESCh. 12.7 - Let G denote the universal gravitational constant...Ch. 12.7 - Suppose that a mass m is in an orbit about a mass...Ch. 12.7 - For a planet in an elliptical orbit about the Sun,...Ch. 12.7 - Suppose that a mass m is an orbit about a mass M...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - Prob. 7ESCh. 12.7 - Prob. 8ESCh. 12.7 - Prob. 9ESCh. 12.7 - Prob. 10ESCh. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - Prob. 12ESCh. 12.7 - Prob. 13ESCh. 12.7 - Prob. 14ESCh. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - Prob. 18ESCh. 12.7 - In exercises that require numerical values, use...Ch. 12.7 - In exercises that require numerical values, use...Ch. 12 - Prob. 1RECh. 12 - Describe the graph of the equation. r=23ti4tjCh. 12 - Describe the graph of the equation....Ch. 12 - Describe the graph of the equation....Ch. 12 - Describe the graph of the equation. r=2i+tj+t21kCh. 12 - Prob. 6RECh. 12 - Find parametric equations for the intersection of...Ch. 12 - In words, give a geometric description of the...Ch. 12 - Find parametric equations of the line tangent to...Ch. 12 - Evaluate costi+sintjdt.Ch. 12 - Evaluate 0/3cos3t,sin3tdt.Ch. 12 - Solve the vector initial-value problem...Ch. 12 - Prob. 17RECh. 12 - Find the arc length parametrization of the line...Ch. 12 - Find an arc length parametrization of the curve...Ch. 12 - Prob. 21RECh. 12 - State the definition of "curvature" and explain...Ch. 12 - Find the curvature of the curve at the stated...Ch. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Suppose that rt is the position function of a...Ch. 12 - (a) What does Theorem 12.2.8 tell you about the...Ch. 12 - As illustrated in the accompanying figure on the...Ch. 12 - If a particle of mass m has uniform circular...Ch. 12 - At time t=0 a particle at the origin of an...Ch. 12 - Prob. 34RECh. 12 - Use Formula (23) in Section 12.7 and refer to...Ch. 12 - As illustrated in the accompanying figure, the...Ch. 12 - A player throws a ball with an initial speed of 60...Ch. 12 - Prob. 1MCCh. 12 - Prob. 2MCCh. 12 - Prob. 3MCCh. 12 - Prob. 5MCCh. 12 - Suppose that the position function of a point...
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ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
Derivatives of Trigonometric Functions - Product Rule Quotient & Chain Rule - Calculus Tutorial; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=_niP0JaOgHY;License: Standard YouTube License, CC-BY