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15.6%20Ex%2036%20-%2041

15.6%20Ex%2036%20-%2041 - S E C T I O N 15.6 The Chain...

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S E C T I O N 15.6 The Chain Rule (ET Section 14.6) 719 Equalities (1) and (2) imply that: u t = − c u x or u t + c u x = 0 35. Jessica and Matthew are running toward the point P along the straight paths that make a fixed angle of θ (Figure 3). Suppose that Matthew runs with velocity v a m / s and Jessica with velocity v b m / s. Let f ( x , y ) be the distance from Matthew to Jessica when Matthew is x meters from P and Jessica is y meters from P . (a) Show that f ( x , y ) = x 2 + y 2 2 xy cos θ . (b) Assume that θ = π / 3. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica is changing when x = 30, y = 20, v a = 4 m / s, and v b = 3 m / s. A B x v a v b y P FIGURE 3 SOLUTION (a) This is a simple application of the Law of Cosines. Connect points A and B in the diagram to form a line segment that we will call f . Then, the Law of Cosines says that f 2 = x 2 + y 2 2 xy cos θ . By taking square roots, we find that f = x 2 + y 2 2 xy cos θ . (b) Using the chain rule, d f dt = f x dx dt + f y dy dt so we get d f dt = ( x y cos θ ) dx / dt x 2 + y 2 2 xy cos θ + ( y x cos θ ) dy / dt x 2 + y 2 2 xy cos θ and using x = 30, y = 20, and dx / dt = 4, dy / dt = 3, we get d f dt = 180 170 cos θ 1300 1200 cos θ Further Insights and Challenges 36. The Law of Cosines states that c 2 = a 2 + b 2 2 ab cos θ , where a , b , c are the sides of a triangle and θ is the angle opposite the side of length c . (a) Use implicit differentiation to compute the derivatives θ a , θ b , and θ c . (b) Suppose that a = 10, b = 16, c = 22. Estimate the change in θ if a and b are increased by 1 and c is increased by 2. SOLUTION (a) Let F ( a , b , c , θ ) = a 2 + b 2 2 ab cos θ c 2 . We use the formulas obtained by implicit differentiation (Eq. (6)) to write θ a = − F a F θ , θ b = − F b F θ , θ c = − F c F θ (1) The partial derivatives of F are F a = 2 a 2 b cos θ , F b = 2 b 2 a cos θ , F c = − 2 c , F θ = 2 ab sin θ Substituting these derivatives in (1), we obtain θ a = − 2 a 2 b cos θ 2
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