To determine Show that the ellipse x = a cos t , y = b sin t , a > b > 0 , has its largest curvature on its major axis and its smallest curvature on its minor axis. Explanation Given information: x = a cos t , y = b sin t , a > b > 0 Formula used: Write the general expression to calculate the velocity. v = d r d t (1) Here, d r d t is the rate of change in displacement with respect to time. Calculation: The vector function is expressed as follows, r = x i + y j + z k = ( a cos t ) i + ( b sin t ) j + ( 0 ) k r = ( a cos t ) i + ( b sin t ) j (2) Substitute equation (2) in equation (1). v = d d t ( ( a cos t ) i + ( b sin t ) j ) = a ( − sin t ) i + b ( cos t ) j = ( − a sin t ) i + ( b cos t ) j The acceleration can be calculated as follows, a = d v d t Substitute ( − a sin t ) i + ( b cos t ) j for v in above equation to find a . a = d d t ( ( − a sin t ) i + ( b cos t ) j ) = − a ( cos t ) i + b ( − sin t ) j = ( − a cos t ) i + ( − b sin t ) j Find the cross product of velocity and acceleration. v × a = | i j k − a sin t b cos t 0 − a cos t − b sin t 0 | = ( 0 − 0 ) i − ( 0 − 0 ) j + ( a b sin 2 t + a b cos 2 t ) k = 0 i − 0 j + ( a b ( sin 2 t + cos 2 t ) ) k = a b k Take magnitude for the above equation. | v × a | = | a b | = a b The curvature can be computed as follows, κ ( t ) = | v × a | | v | 3 Substitute a b for | v × a | and ( − a sin t ) i + ( b cos t ) j for v in above equation to find κ ( t )

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

4th Edition

ISBN: 9780135164860

Author: Joel R. Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 12.4, Problem 18E

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Show that the ellipse x=acost, y=bsint, a>b>0, has its largest curvature on its major axis and its smallest curvature on its minor axis.

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Chapter 12.4, Problem 17E

Chapter 12.4, Problem 19E

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Write the given third order linear equation as an equivalent system of first order equations with initial values. Use Y1 = Y, Y2 = y', and y3 = y". - - √ (3t¹ + 3 − t³)y" — y" + (3t² + 3)y' + (3t — 3t¹) y = 1 − 3t² \y(3) = 1, y′(3) = −2, y″(3) = −3 (8) - (888) - with initial values Y = If you don't get this in 3 tries, you can get a hint.

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Chapter 12 Solutions

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

Ch. 12.1 - In Exercises 1–4, find the given limits. 1. Ch. 12.1 - In Exercises 1–4, find the given limits. 2. Ch. 12.1 - In Exercises 1–4, find the given limits. 3. Ch. 12.1 - In Exercises 1–4, find the given limits. 4. Ch. 12.1 - Motion in the Plane In Exercises 58, r(t) is the...Ch. 12.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 12.1 - In Exercises 58, r(t) is the position of a...Ch. 12.1 - In Exercises 5–8, r(t) is the position of a...Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E

Ch. 12.1 - Exercises 9–12 give the position vectors of...Ch. 12.1 - Prob. 12E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 14E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - In Exercises 1922, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - Prob. 22E Ch. 12.1 - As mentioned in the text, the tangent line to a...Ch. 12.1 - Prob. 24E Ch. 12.1 - Tangents to Curves As mentioned in the text, the...Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Prob. 33E Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Motion along a circle Each of the following...Ch. 12.1 - Motion along a circle Show that the vector-valued...Ch. 12.1 - Prob. 39E Ch. 12.1 - Motion along a cycloid A particle moves in the...Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Component test for continuity at a point Show that...Ch. 12.1 - Limits of cross products of vector functions...Ch. 12.1 - Differentiable vector functions are continuous...Ch. 12.1 - Constant Function Rule Prove that if u is the...Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 1. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 2. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 3. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 4. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 5. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 6. Ch. 12.2 - Evaluate the integrals in Exercises 110. 7....Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 8. Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Prob. 15E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Prob. 18E Ch. 12.2 - Prob. 19E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - At time t = 0, a particle is located at the point...Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Range and height versus speed Show that doubling a...Ch. 12.2 - Flight time and height A projectile is fired with...Ch. 12.2 - Prob. 26E Ch. 12.2 - Prob. 27E Ch. 12.2 - Beaming electrons An electron in a TV tube is...Ch. 12.2 - Prob. 29E Ch. 12.2 - Finding muzzle speed Find the muzzle speed of a...Ch. 12.2 - Prob. 31E Ch. 12.2 - Colliding marbles The accompanying figure shows an...Ch. 12.2 - Firing from (x0, y0) Derive the equations (see...Ch. 12.2 - Where trajectories crest For a projectile fired...Ch. 12.2 - Prob. 35E Ch. 12.2 - Prob. 36E Ch. 12.2 - Prob. 37E Ch. 12.2 - Products of scalar and vector functions Suppose...Ch. 12.2 - Prob. 39E Ch. 12.2 - The Fundamental Theorem of Calculus The...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - Prob. 7E Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - Find the point on the curve at a distance 26...Ch. 12.3 - Find the point on the curve at a distance 13...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - Arc length Find the length of the curve from (0,...Ch. 12.3 - Length of helix The length of the turn of the...Ch. 12.3 - Prob. 17E Ch. 12.3 - Length is independent of parametrization To...Ch. 12.3 - The involute of a circle If a siring wound around...Ch. 12.3 - Prob. 20E Ch. 12.3 - Distance along a line Show that if u is a unit...Ch. 12.3 - Prob. 22E Ch. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Find T, N, and for the plane curves in Exercises...Ch. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Prob. 5E Ch. 12.4 - Prob. 6E Ch. 12.4 - Prob. 7E Ch. 12.4 - Prob. 8E Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Prob. 16E Ch. 12.4 - Show that the parabola , has its largest curvature...Ch. 12.4 - Show that the ellipse x = a cos t, y = b sin t, a...Ch. 12.4 - Prob. 19E Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.4 - Prob. 22E Ch. 12.4 - Prob. 23E Ch. 12.4 - Prob. 24E Ch. 12.4 - Prob. 25E Ch. 12.4 - Prob. 26E Ch. 12.4 - Prob. 27E Ch. 12.4 - Prob. 28E Ch. 12.4 - Prob. 29E Ch. 12.4 - Prob. 30E Ch. 12.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 12.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 12.5 - In Exercises 36, write a in the form a = aTT + aNN...Ch. 12.5 - Prob. 4E Ch. 12.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 12.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 12.5 - In Exercises 7 and 8, find r, T, N, and B at the...Ch. 12.5 - Prob. 8E Ch. 12.5 - The speedometer on your car reads a steady 35 mph....Ch. 12.5 - Prob. 10E Ch. 12.5 - Can anything be said about the speed of a particle...Ch. 12.5 - An object of mass m travels along the parabola y =...Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - Prob. 15E Ch. 12.5 - Prob. 16E Ch. 12.6 - Prob. 1E Ch. 12.6 - Prob. 2E Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - Prob. 5E Ch. 12.6 - Prob. 6E Ch. 12.6 - Prob. 7E Ch. 12.6 - Prob. 8E Ch. 12.6 - Prob. 9E Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 11E Ch. 12.6 - Prob. 12E Ch. 12.6 - Prob. 13E Ch. 12.6 - Prob. 14E Ch. 12.6 - Prob. 15E Ch. 12.6 - Prob. 16E Ch. 12.6 - Prob. 17E Ch. 12.6 - Prob. 18E Ch. 12 - Prob. 1GYR Ch. 12 - Prob. 2GYR Ch. 12 - Prob. 3GYR Ch. 12 - Prob. 4GYR Ch. 12 - Prob. 5GYR Ch. 12 - Prob. 6GYR Ch. 12 - Prob. 7GYR Ch. 12 - Prob. 8GYR Ch. 12 - Prob. 9GYR Ch. 12 - Prob. 10GYR Ch. 12 - Prob. 11GYR Ch. 12 - Prob. 12GYR Ch. 12 - Prob. 13GYR Ch. 12 - In Exercises 1 and 2, graph the curves and sketch...Ch. 12 - Prob. 2PE Ch. 12 - Prob. 3PE Ch. 12 - Prob. 4PE Ch. 12 - Prob. 5PE Ch. 12 - Prob. 6PE Ch. 12 - Prob. 7PE Ch. 12 - Prob. 8PE Ch. 12 - Prob. 9PE Ch. 12 - Prob. 10PE Ch. 12 - Prob. 11PE Ch. 12 - Prob. 12PE Ch. 12 - Prob. 13PE Ch. 12 - Prob. 14PE Ch. 12 - Prob. 15PE Ch. 12 - Prob. 16PE Ch. 12 - Prob. 17PE Ch. 12 - Prob. 18PE Ch. 12 - Prob. 19PE Ch. 12 - In Exercises 17-20, find T, N, B, and k at the...Ch. 12 - Prob. 21PE Ch. 12 - Prob. 22PE Ch. 12 - Prob. 23PE Ch. 12 - Prob. 24PE Ch. 12 - Prob. 25PE Ch. 12 - Find equations for the osculating, normal, and...Ch. 12 - Find parametric equations for the line that is...Ch. 12 - Prob. 28PE Ch. 12 - Prob. 29PE Ch. 12 - Prob. 30PE Ch. 12 - Prob. 1AAE Ch. 12 - Suppose the curve in Exercise 1 is replaced by the...Ch. 12 - Prob. 3AAE Ch. 12 - Prob. 4AAE Ch. 12 - Prob. 5AAE Ch. 12 - Prob. 6AAE Ch. 12 - Prob. 7AAE Ch. 12 - Prob. 8AAE

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Show that the ellipse x = a cos t , y = b sin t , a &gt; b &gt; 0 , has its largest curvature on its major axis and its smallest curvature on its minor axis.

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Show that the ellipse x=acost, y=bsint, a>b>0, has its largest curvature on its major axis and its smallest curvature on its minor axis.

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Chapter 12 Solutions

Show that the ellipse x = a cos t , y = b sin t , a > b > 0 , has its largest curvature on its major axis and its smallest curvature on its minor axis.