Thomas' Calculus - With MyMathLab
14th Edition
ISBN: 9780134665672
Author: Hass
Publisher: PEARSON
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Chapter 13, Problem 26PE
To determine
Find the equations for the osculating, normal, and rectifying planes for given
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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
Thomas' Calculus - With MyMathLab
Ch. 13.1 - In Exercises 1–4, find the given limits.
1.
Ch. 13.1 - In Exercises 1–4, find the given limits.
2.
Ch. 13.1 - In Exercises 1–4, find the given limits.
3.
Ch. 13.1 - In Exercises 1–4, find the given limits.
4.
Ch. 13.1 - Motion in the Plane
In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane
In Exercises 5–8, r(t) is the...Ch. 13.1 - In Exercises 5–8, r(t) is the position of a...Ch. 13.1 - In Exercises 5–8, r(t) is the position of a...Ch. 13.1 - Prob. 9ECh. 13.1 - Prob. 10E
Ch. 13.1 - Exercises 9–12 give the position vectors of...Ch. 13.1 - Prob. 12ECh. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 13–18, r(t) is the position of a...Ch. 13.1 - In Exercises 19–22, r(t) is the position of a...Ch. 13.1 - In Exercises 19–22, r(t) is the position of a...Ch. 13.1 - In Exercises 19–22, r(t) is the position of a...Ch. 13.1 - Prob. 22ECh. 13.1 - As mentioned in the text, the tangent line to a...Ch. 13.1 - Tangents to Curves
As mentioned in the text, the...Ch. 13.1 - Tangents to Curves
As mentioned in the text, the...Ch. 13.1 - Tangents to Curves
As mentioned in the text, the...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - In Exercises 31–36, r(t) is the position of a...Ch. 13.1 - In Exercises 31–36, r(t) is the position of a...Ch. 13.1 - In Exercises 31–36, r(t) is the position of a...Ch. 13.1 - In Exercises 31–36, r(t) is the position of a...Ch. 13.1 - Prob. 35ECh. 13.1 - In Exercises 31–36, r(t) is the position of a...Ch. 13.1 - Motion along a circle Each of the following...Ch. 13.1 - Motion along a circle Show that the vector-valued...Ch. 13.1 - Motion along a parabola A particle moves along the...Ch. 13.1 - Motion along a cycloid A particle moves in the...Ch. 13.1 - Let r be a differentiable vector function of t....Ch. 13.1 - Prob. 42ECh. 13.1 - Prob. 43ECh. 13.1 - Prob. 44ECh. 13.1 - Prob. 45ECh. 13.1 - Limits of cross products of vector functions...Ch. 13.1 - Differentiable vector functions are continuous...Ch. 13.1 - Constant Function Rule Prove that if u is the...Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
1.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
2.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
3.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
4.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
5.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
6.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
7.
Ch. 13.2 - Prob. 8ECh. 13.2 - Evaluate the integrals in Exercises 1–10.
9.
Ch. 13.2 - Evaluate the integrals in Exercises 1–10.
10.
Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - At time t = 0, a particle is located at the point...Ch. 13.2 - Prob. 22ECh. 13.2 - Travel time A projectile is fired at a speed of...Ch. 13.2 - Range and height versus speed
Show that doubling a...Ch. 13.2 - Flight time and height A projectile is fired with...Ch. 13.2 - Throwing a baseball A baseball is thrown from the...Ch. 13.2 - Firing golf balls A spring gun at ground level...Ch. 13.2 - Prob. 28ECh. 13.2 - Equal-range firing angles What two angles of...Ch. 13.2 - Prob. 30ECh. 13.2 - Prob. 31ECh. 13.2 - Colliding marbles The accompanying figure shows an...Ch. 13.2 - Firing from (x0, y0) Derive the equations
(see...Ch. 13.2 - Where trajectories crest For a projectile fired...Ch. 13.2 -
Launching downhill An ideal projectile is...Ch. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - Prob. 39ECh. 13.2 - The view from Skylab 4 What percentage of Earth’s...Ch. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Hitting a baseball with linear drag Consider the...Ch. 13.2 - Prob. 43ECh. 13.2 - Products of scalar and vector functions Suppose...Ch. 13.2 - Antiderivatives of vector functions
Use Corollary...Ch. 13.2 - The Fundamental Theorem of Calculus The...Ch. 13.2 -
Hitting a baseball with linear drag under a wind...Ch. 13.2 - Prob. 48ECh. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - Find the point on the curve
at a distance 26...Ch. 13.3 -
Find the point on the curve
r(t) = (12 sin t)i −...Ch. 13.3 - In Exercises 11–14, find the arc length parameter...Ch. 13.3 - In Exercises 11–14, find the arc length parameter...Ch. 13.3 - In Exercises 11–14, find the arc length parameter...Ch. 13.3 - In Exercises 11–14, find the arc length parameter...Ch. 13.3 - Arc length Find the length of the curve
from (0,...Ch. 13.3 - Length of helix The length of the turn of the...Ch. 13.3 - Length is independent of parametrization To...Ch. 13.3 - The involute of a circle If a siring wound around...Ch. 13.3 - (Continuation of Exercise 19.) Find the unit...Ch. 13.3 - Prob. 21ECh. 13.3 - Prob. 22ECh. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - Prob. 3ECh. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - A formula for the curvature of the graph of a...Ch. 13.4 - A formula for the curvature of a parametrized...Ch. 13.4 -
Normals to plane curves
Show that n(t) = −g′(t)i...Ch. 13.4 - (Continuation of Exercise 7.)
Use the method of...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Show that the parabola , has its largest curvature...Ch. 13.4 - Show that the ellipse x = a cos t, y = b sin t, a...Ch. 13.4 - Maximizing the curvature of a helix In Example 5,...Ch. 13.4 - Prob. 20ECh. 13.4 - Find an equation for the circle of curvature of...Ch. 13.4 - Find an equation for the circle of curvature of...Ch. 13.4 - Prob. 23ECh. 13.4 - Prob. 24ECh. 13.4 - Prob. 25ECh. 13.4 - Prob. 26ECh. 13.4 - Prob. 27ECh. 13.4 - Prob. 28ECh. 13.4 - Osculating circle Show that the center of the...Ch. 13.4 - Osculating circle Find a parametrization of the...Ch. 13.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 13.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 13.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 13.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 13.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 13.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 13.5 - In Exercises 7 and 8, find r, T, N, and B at the...Ch. 13.5 - In Exercises 7 and 8, find r, T, N, and B at the...Ch. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - Prob. 10ECh. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - Prob. 14ECh. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - In Exercises 9–16 of Section 13.4, you found T, N,...Ch. 13.5 - Prob. 17ECh. 13.5 - Prob. 18ECh. 13.5 - Prob. 19ECh. 13.5 - Prob. 20ECh. 13.5 - Prob. 21ECh. 13.5 - Prob. 22ECh. 13.5 - A sometime shortcut to curvature If you already...Ch. 13.5 - What can be said about the torsion of a smooth...Ch. 13.5 - Differentiable curves with zero torsion lie in...Ch. 13.5 - A formula that calculates τ from B and v If we...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - Prob. 7ECh. 13.6 - Prob. 8ECh. 13.6 - Circular orbits Show that a planet in a circular...Ch. 13.6 - Prob. 10ECh. 13.6 - Prob. 11ECh. 13.6 - Do the data in the accompanying table support...Ch. 13.6 - Prob. 13ECh. 13.6 - Prob. 14ECh. 13.6 - Prob. 15ECh. 13.6 - Prob. 16ECh. 13.6 - Prob. 17ECh. 13.6 - Prob. 18ECh. 13 - Prob. 1GYRCh. 13 - How do you define and calculate the velocity,...Ch. 13 - Prob. 3GYRCh. 13 - Prob. 4GYRCh. 13 - Prob. 5GYRCh. 13 - Prob. 6GYRCh. 13 - Prob. 7GYRCh. 13 - Define curvature, circle of curvature (osculating...Ch. 13 - Prob. 9GYRCh. 13 - Prob. 10GYRCh. 13 - Prob. 11GYRCh. 13 - Prob. 12GYRCh. 13 - Prob. 13GYRCh. 13 - In Exercises 1 and 2, graph the curves and sketch...Ch. 13 - Prob. 2PECh. 13 - Prob. 3PECh. 13 - Prob. 4PECh. 13 - Finding curvature At point P, the velocity and...Ch. 13 - Prob. 6PECh. 13 - Prob. 7PECh. 13 - Prob. 8PECh. 13 - Prob. 9PECh. 13 - Speed along a cycloid A circular wheel with radius...Ch. 13 - Prob. 11PECh. 13 - Javelin A javelin leaves the thrower’s hand 7 ft...Ch. 13 - Prob. 13PECh. 13 - Javelin In Potsdam in 1988, Petra Felke of (then)...Ch. 13 - Prob. 15PECh. 13 - Find the lengths of the curves in Exercises 15 and...Ch. 13 - Prob. 17PECh. 13 - Prob. 18PECh. 13 - In Exercises 17-20, find T, N, B, and k at the...Ch. 13 - Prob. 20PECh. 13 - In Exercises 21 and 22, write a in the form a =...Ch. 13 - Prob. 22PECh. 13 - Prob. 23PECh. 13 - Prob. 24PECh. 13 - Prob. 25PECh. 13 - Prob. 26PECh. 13 - Find parametric equations for the line that is...Ch. 13 - Find parametric equations for the line that is...Ch. 13 - Prob. 29PECh. 13 - Prob. 30PECh. 13 - Prob. 31PECh. 13 - The view from Skylab 4 What percentage of Earth’s...Ch. 13 - Prob. 1AAECh. 13 - Suppose the curve in Exercise 1 is replaced by the...Ch. 13 - Prob. 3AAECh. 13 - Prob. 4AAECh. 13 - Prob. 5AAECh. 13 - Express the curvature of a twice-differentiable...Ch. 13 - Prob. 7AAECh. 13 - Prob. 8AAECh. 13 - Unit vectors for position and motion in...
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