Multivariable Calculus
8th Edition
ISBN: 9781305266643
Author: James Stewart
Publisher: Cengage Learning
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Chapter 13.1, Problem 7E
To determine
To sketch: the curve of
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Exercise 11.3 A slope field is given for the equation y' = 4y+4.
(a) Sketch the particular solution that corresponds to y(0) = −2
(b) Find the constant solution
(c) For what initial conditions y(0) is the solution increasing?
(d) For what initial conditions y(0) is the solution decreasing?
(e) Verify these results using only the differential equation y' = 4y+4.
Chapter 13 Solutions
Multivariable Calculus
Ch. 13.1 - Prob. 1ECh. 13.1 - Prob. 2ECh. 13.1 - Find the limit. 3. limt0(e3ti+t2sin2tj+cos2tk)Ch. 13.1 - Find the limit. 4. limt1(t2-tt-1i+t+8j+sintlntk)Ch. 13.1 - Find the limit. 5. limt1+t21t2,tan-1t,1e2ttCh. 13.1 - Prob. 6ECh. 13.1 - Prob. 7ECh. 13.1 - Prob. 8ECh. 13.1 - Prob. 9ECh. 13.1 - Prob. 10E
Ch. 13.1 - Prob. 11ECh. 13.1 - Prob. 12ECh. 13.1 - Prob. 13ECh. 13.1 - Prob. 14ECh. 13.1 - Prob. 15ECh. 13.1 - Prob. 16ECh. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Prob. 18ECh. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Prob. 20ECh. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Prob. 27ECh. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Prob. 29ECh. 13.1 - Prob. 30ECh. 13.1 - Prob. 31ECh. 13.1 - At what points does the helix r(t) = sin t, cos t,...Ch. 13.1 - Graph the curve with parametric equations x = sin...Ch. 13.1 - Graph the curve with parametric equations x = (1 +...Ch. 13.1 - Prob. 40ECh. 13.1 - Prob. 41ECh. 13.1 - Prob. 42ECh. 13.1 - Prob. 43ECh. 13.1 - Prob. 44ECh. 13.1 - Prob. 45ECh. 13.1 - Prob. 46ECh. 13.1 - If two objects travel through space along two...Ch. 13.1 - Prob. 50ECh. 13.1 - Prob. 53ECh. 13.2 - Prob. 1ECh. 13.2 - Prob. 2ECh. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Prob. 4ECh. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Prob. 6ECh. 13.2 - Prob. 7ECh. 13.2 - Prob. 8ECh. 13.2 - Find the derivative of the vector function. 9....Ch. 13.2 - Prob. 10ECh. 13.2 - Prob. 11ECh. 13.2 - Prob. 12ECh. 13.2 - Prob. 13ECh. 13.2 - Prob. 14ECh. 13.2 - Prob. 15ECh. 13.2 - Prob. 16ECh. 13.2 - Prob. 17ECh. 13.2 - Prob. 18ECh. 13.2 - Prob. 19ECh. 13.2 - Prob. 20ECh. 13.2 - If r(t) = t, t2, t3, find r'(t), T( 1), r"(t). and...Ch. 13.2 - Prob. 22ECh. 13.2 - Prob. 23ECh. 13.2 - Prob. 24ECh. 13.2 - Prob. 25ECh. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find a vector equation for the tangent line to the...Ch. 13.2 - Find the point on the curve r(t) = 2 cos t, 2 sin...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Prob. 32ECh. 13.2 - Prob. 33ECh. 13.2 - At what point do the curves r1(t) = t, 1 t, 3 +...Ch. 13.2 - Evaluate the integral. 35. 02(ti-t3j+3t5k)dtCh. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Evaluate the integral. 38....Ch. 13.2 - Evaluate the integral. 39....Ch. 13.2 - Evaluate the integral. 40. (te2ti+t1-tj+11-t2k)dtCh. 13.2 - Find r(t) if r'(t) = 2t i + 3t2 j + t k and r(1) =...Ch. 13.2 - Prob. 42ECh. 13.2 - Prove Formula 1 of Theorem 3.Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prob. 45ECh. 13.2 - Prove Formula 6 of Theorem 3.Ch. 13.2 - Prob. 47ECh. 13.2 - Prob. 48ECh. 13.2 - Find f'(2), where f(t) = u(t) v(t), u(2) = 1, 2,...Ch. 13.2 - If r(t) = u(t) v(t), where u and v are the vector...Ch. 13.2 - If r(t) = a cos t + b sin t, where a and b are...Ch. 13.2 - If r is the vector function in Exercise 51, show...Ch. 13.2 - Show that if r is a vector function such that r''...Ch. 13.2 - Prob. 54ECh. 13.2 - If r(t) 0, show that ddtr(t)=1r(t)r(t)r(t)....Ch. 13.2 - Prob. 56ECh. 13.2 - Prob. 57ECh. 13.2 - Prob. 58ECh. 13.3 - Find the length of the curve. 1. r(t) =t, 3 cos t,...Ch. 13.3 - Find the length of the curve. 2. r(t)=2t,t2,13t3,...Ch. 13.3 - Prob. 3ECh. 13.3 - Find the length of the curve. 4. r(t) =cos t i +...Ch. 13.3 - Find the length of the curve. 5. r(t) = i + t2 j +...Ch. 13.3 - Find the length of the curve. 6. r(t) = t2 i + 9t...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Prob. 9ECh. 13.3 - Graph the curve with parametric equations x = sin...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Prob. 12ECh. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - Prob. 14ECh. 13.3 - Suppose you start at the point (0, 0. 3) and move...Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 18ECh. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 20ECh. 13.3 - Use Theorem 10 to find the curvature. 21. r(t) =...Ch. 13.3 - Use Theorem 10 to find the curvature. 22. r(t) = t...Ch. 13.3 - Prob. 23ECh. 13.3 - Prob. 24ECh. 13.3 - Find the curvature of r(t) = t, t2, t3 at the...Ch. 13.3 - Graph the curve with parametric equations x = cos...Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - Prob. 28ECh. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - Prob. 31ECh. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - (a) Is the curvature of the curve C shown in the...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Prob. 39ECh. 13.3 - Prob. 42ECh. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Consider the curvature at x = 0 for each member of...Ch. 13.3 - Prob. 47ECh. 13.3 - Prob. 48ECh. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - At what point on the curve x = t3, y = 3t, z = t4...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Prob. 56ECh. 13.3 - Prob. 58ECh. 13.3 - Prob. 59ECh. 13.3 - Show that the curvature of a plane curve is =...Ch. 13.3 - Prob. 62ECh. 13.3 - Prob. 63ECh. 13.3 - Show that the circular helix r(t) = a cos t, a sin...Ch. 13.3 - Prob. 65ECh. 13.3 - Find the curvature and torsion of the curve x =...Ch. 13.3 - The DNA molecule has the shape of a double helix...Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Prob. 5ECh. 13.4 - Prob. 6ECh. 13.4 - Prob. 7ECh. 13.4 - Prob. 8ECh. 13.4 - Prob. 9ECh. 13.4 - Prob. 10ECh. 13.4 - Prob. 11ECh. 13.4 - Prob. 12ECh. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 14ECh. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - Prob. 16ECh. 13.4 - The position function of a particle is given by...Ch. 13.4 - Prob. 20ECh. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Rework Exercise 23 if the projectile is fired from...Ch. 13.4 - A ball is thrown at an angle of 45 to the ground....Ch. 13.4 - A projectile is tired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - Prob. 28ECh. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Prob. 30ECh. 13.4 - Prob. 31ECh. 13.4 - A ball with mass 0.8 kg is thrown southward into...Ch. 13.4 - Prob. 34ECh. 13.4 - Prob. 35ECh. 13.4 - (a) If a particle moves along a straight line,...Ch. 13.4 - Prob. 37ECh. 13.4 - Prob. 38ECh. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Prob. 40ECh. 13.4 - Prob. 41ECh. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - If a particle with mass m moves with position...Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - A rocket burning its onboard fuel while moving...Ch. 13 - Prob. 1RCCCh. 13 - What is the connection between vector functions...Ch. 13 - Prob. 3RCCCh. 13 - Prob. 4RCCCh. 13 - Prob. 5RCCCh. 13 - Prob. 6RCCCh. 13 - Prob. 7RCCCh. 13 - Prob. 8RCCCh. 13 - Prob. 9RCCCh. 13 - Determine whether the statement is true or false....Ch. 13 - Prob. 2RQCh. 13 - Prob. 3RQCh. 13 - Prob. 4RQCh. 13 - Prob. 5RQCh. 13 - Prob. 6RQCh. 13 - Prob. 7RQCh. 13 - Prob. 8RQCh. 13 - Prob. 9RQCh. 13 - Determine whether the statement is true or false....Ch. 13 - Prob. 11RQCh. 13 - Prob. 12RQCh. 13 - Prob. 13RQCh. 13 - Prob. 14RQCh. 13 - (a) Sketch the curve with vector function r(t) = t...Ch. 13 - Let r(t) = 2-t, (et 1)/t, ln(t + 1). (a) Find the...Ch. 13 - Prob. 3RECh. 13 - Find parametric equations for the tangent line to...Ch. 13 - If r(t) = t2 i + t cos t j + sin t k, evaluate...Ch. 13 - Let C be the curve with equations x = 2 t3 y = 2t...Ch. 13 - Use Simpsons Rule with n = 6 to estimate the...Ch. 13 - Prob. 8RECh. 13 - Prob. 9RECh. 13 - Prob. 10RECh. 13 - Prob. 11RECh. 13 - Prob. 12RECh. 13 - Find the curvature of the curve y = x4 at the...Ch. 13 - Find an equation of the osculating circle of the...Ch. 13 - Prob. 15RECh. 13 - The figure shows the curve C traced by a particle...Ch. 13 - A particle moves with position function r(t) = t...Ch. 13 - Find the velocity, speed, and acceleration of a...Ch. 13 - A particle starts at the origin with initial...Ch. 13 - An athlete throws a shot at an angle of 45 to the...Ch. 13 - A projectile is launched with an initial speed of...Ch. 13 - Prob. 22RECh. 13 - Prob. 23RECh. 13 - PROBLEM PLUS FIGURE FOR PROBLEM 1 1. A particle P...Ch. 13 - Prob. 2PCh. 13 - A projectile is fired from the origin with angle...Ch. 13 - (a) A projectile it fired from the origin down an...Ch. 13 - Prob. 5PCh. 13 - Prob. 6PCh. 13 - If a projectile is fired with angle of elevation ...Ch. 13 - A cable has radius r and length L and is wound...Ch. 13 - Prob. 9P
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