To determine Calculate the vector r from the given initial value problem. Explanation Given information: Consider that the differential equation. d r d t = ( t t 2 + 2 ) i − ( t 2 + 1 t − 2 ) j + ( t 2 + 4 t 2 + 3 ) k , t < 2 (1) Consider that the initial condition. r ( 0 ) = i − j + k (2) Calculation: Integrate both sides of Equation (1) to find the vector r . ∫ d r d t = ∫ [ ( t t 2 + 2 ) i − ( t 2 + 1 t − 2 ) j + ( t 2 + 4 t 2 + 3 ) k ] d t = ∫ [ ( 1 2 2 t t 2 + 2 ) i − ( t + 2 + 5 t − 2 ) j + ( 1 + 1 t 2 + 3 ) k ] d t r = [ 1 2 ln ( t 2 + 2 ) ] i − [ 1 2 t 2 + 2 t + 5 ln | t − 2 | ] j + [ t + 1 3 tan − 1 t 3 ] k + C (3) Substitute 0 for t in Equation (3). r ( 0 ) = [ 1 2 ln ( 0 2 + 2 ) ] i − [ 1 2 0 2 + 2 ( 0 ) + 5 ln | 0 − 2 | ] j + [ 0 + 1 3 tan − 1 0 3 ] k + C r ( 0 ) = ( 1 2 ln 2 ) i − ( 5 ln 2 ) j + C (4) Compare Equations (2) and (4)

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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Differential Equations

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Chapter 12.2, Problem 16E

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Calculate the vector r from the given initial value problem.

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A factorization A = PDP 1 is not unique. For A= 7 2 -4 1 1 1 5 0 2 1 one factorization is P = D= and P-1 30 = Use this information with D₁ = to find a matrix P₁ such that - -1 -2 0 3 1 - - 1 05 A-P,D,P P1 (Type an integer or simplified fraction for each matrix element.)

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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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