Explanation Given information: The distance traveled by an object is 10 m. Calculation: Split the movement of an object as three parts. Draw the separated portion in the graph as in Figure (1). Refer Figure (1), Determine the work done by each force using the calculation of area beneath the graph. W = A 1 + A 2 + A 3 = 1 4 π × F 1 × d 1 + F 2 d 2 2 + F 3 d 3 + F 4 d 4 2 Here, A 1 is the area of portion 1, d 1 is the radius of the quarter circle, F 1 is the force an object in portion 1, A 2 is the area of portion 2, F 2 is the force an object in portion 2, d 2 is the distance moved by an object in portion 2, A 3 is the area of portion 3, F 3 is the force an object in portion 3, d 3 is the distance moved by an object in portion 3, F 4 is the force an object in portion 4, and d 4 is the distance moved by an object in portion 4

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ISBN: 9780134654874

Author: WEIR

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Chapter 6.5, Problem 2E

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Find the work done by each force in moving an object for 10 m distance.

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Chapter 6.5, Problem 1E

Chapter 6.5, Problem 3E

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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.

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Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.

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