15.6%20Ex%2036%20-%2041

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

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SECTION 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 fxed angle oF θ (±igure 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 ) = p x 2 + y 2 2 xy cos . (b) Assume that = 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 = 4m / s, and v b = 3m / 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 cos . By taking square roots, we fnd that f = p x 2 + y 2 2 cos . (b) Using the chain rule, df dt = f x dx + f y dy so we get = ( x y cos ) / p x 2 + y 2 2 cos + ( y x cos ) / p x 2 + y 2 2 cos and using x = 30, y = 20, and / = 4, / = 3, we get = 180 170 cos 1300 1200 cos Further Insights and Challenges 36. The Law oF Cosines states that c 2 = a 2 + b 2 2 ab cos ,where a , b , c are the sides oF a triangle and is the angle opposite the side oF length c . (a) Use implicit diFFerentiation to compute the derivatives a , b ,and c . (b) Suppose that a = 10, b = 16, c = 22. Estimate the change in iF a and b are increased by 1 and c is increased by 2. (a) Let F ( a , b , c , ) = a 2 + b 2 2 cos c 2 . We use the Formulas obtained by implicit diFFerentiation (Eq. (6)) to write a F a F , b F b F , c F c F (1) The partial derivatives oF F are F a = 2 a 2 b cos , F b = 2 b 2 a cos , F c 2 c , F = 2 sin Substituting these derivatives in (1), we obtain a 2 a 2 b cos 2 sin a b cos sin b 2 b 2 a cos 2 sin b a cos sin
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720 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) θ c =− 2 c 2 ab sin = c sin (b) The linear approximation for is 1
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15.6%20Ex%2036%20-%2041 - S E C T I O N 15.6 The Chain Rule...

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