Explanation Given information: f ( x ) = x x + 1 for x > − 1 (1) Formula used: Write a general expression to calculate the maximum curvature. κ ( x ) = | f ″ ( x ) | [ 1 + ( f ′ ( x ) ) 2 ] 3 2 (2) Here, f ′ ( x ) is the first order derivative, and f ″ ( x ) is second order derivative. Calculation: Take first order derivative for equation (1). f ′ ( x ) = ( 1 ) ( x + 1 ) − x ( 1 + 0 ) ( x + 1 ) 2 = x + 1 − x ( x + 1 ) 2 = 1 ( x + 1 ) 2 Take a derivative for above equation. f ″ ( x ) = − 2 ( x + 1 ) − 2 − 1 = − 2 ( x + 1 ) − 3 = − 2 ( x + 1 ) 3 Substitute 1 ( x + 1 ) 2 for f ′ ( x ) , and − 2 ( x + 1 ) 3 for f ″ ( x ) in equation (2) to find κ ( x ) . κ ( x ) = | − 2 ( x + 1 ) 3 | [ 1 + ( 1 ( x + 1 ) 2 ) 2 ] 3 2 = 2 ( x + 1 ) 3 [ 1 + ( 1 2 ( ( x + 1 ) 2 ) 2 ) ] 3 2 = 2 ( x + 1 ) 3 [ 1 + ( 1 ( x + 1 ) 4 ) ] 3 2 { ∵ ( x a ) b = x a b } = 2 ( x + 1 ) 3 [ 1 + ( 1 ( x + 1 ) 4 ) ] 3 2 Simplify the above equation to find κ ( x ) . κ ( x ) = 2 ( x + 1 ) 3 [ ( ( x + 1 ) 4 + 1 ( x + 1 ) 4 ) ] 3 2 = 2 ( x + 1 ) 3 [ ( [ ( x + 1 ) 4 + 1 ] 3 2 ( ( x + 1 ) 4 ) 3 2 ) ] { ∵ ( x y ) a = x a y a } = 2 ( x + 1 ) 3 [ ( [ ( x + 1 ) 4 + 1 ] 3 2 ( ( x + 1 ) 3 ) 4 2 ) ] { ∵ ( x a ) b = x a b } = 2 ( x + 1 ) 3 ( [ ( x + 1 ) 4 + 1 ] 3 2 ( ( x + 1 ) 3 ) 2 ) Simplify the above equation to find κ ( x ) . κ ( x ) = 2 ( [ ( x + 1 ) 4 + 1 ] 3 2 ( x + 1 ) 3 ) κ ( x ) = 2 ( x + 1 ) 3 [ ( x + 1 ) 4 + 1 ] 3 2 (3) To find the maximum value of curvature, differentiate the equation (3) and equate to zero. κ ′ ( x ) = 0 (4) Take derivative for equation (3) to find κ ′ ( x )

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

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

Publisher: PEARSON

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Chapter 12.4, Problem 28E

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Chapter 12 Solutions

University Calculus: Early Transcendentals (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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