Quiz 6 Solution - (b Write an equation for each horizontal...

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MA 113 Name: ANSWER KEY Quiz 6 - October 17, 2013 1. Suppose that we have two variable resistors connected in parallel with resistances R 1 and R 2 and measured in ohms (Ω). The total resistance is given by 1 R ( t ) = 1 R 1 ( t ) + 1 R 2 ( t ) . (a) Find R (0) if R 1 (0) = 30 Ω and R 2 (0) = 20 Ω. (b) Suppose that the resistance R 1 is increasing at a rate of 0.25 Ω /min and R 2 is increasing at a rate 0.5 Ω /min at t = 0. What is the rate of change in R at t = 0? SOLUTION: (a) The first part just asks to find R , so use the given formula at t = 0, 1 R = 1 30 + 1 20 = 2 60 + 3 60 = 5 60 . Now R = 60 5 = 12Ω. (1 point) (b) Part (b) uses related rates. We are looking for dR dt at t = 0. Take the derivative of the formula using implicit differentiation. This results in - 1 [ R ( t )] 2 dR dt = - 1 [ R 1 ( t )] 2 dR 1 dt + - 1 [ R 2 ( t )] 2 dR 2 dt , (3 points) and solving for dR dt we get dR dt = [ R ( t )] 2 [ R 1 ( t )] 2 dR 1 dt + [ R ( t )] 2 [ R 2 ( t )] 2 dR 2 dt . (1 point) Using information for t = 0, we obtain [ R (0)] 2 [ R 1 (0)] 2 dR 1 dt + [ R (0)] 2 [ R 2 (0)] 2 dR 2 dt = 12 2 30 2 ( . 25) + 12 2 20 2 ( . 5) = 4 100 + 18 100 = 22 100 = 0 . 22Ω. (1 point) 2. Let the function f ( x ) = 2 x 2 be as given. (a) Find the x value(s) where the tangent line to the function is horizontal.
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Unformatted text preview: (b) Write an equation for each horizontal tangent line in point-slope form. SOLUTION: (a) Horizontal tangent lines imply slope zero of the function. Take f and set it to zero. Now, f ( x ) = d dx e ln(2) x 2 = e ln(2) x 2 ln(2)(2 x ) = 2 x 2 ln(2)(2 x ) by the chain rule and differentiation of exponential functions, and 2 x 2 ln(2)(2 x ) = 0 when (2 x ) = 0. Thus, x = 0 is the only value of x so that the tangent line is horizontal. (2 point) (b) Writing this tangent line in point slope form requires a point, ( x ,y ) and a slope m , and the general equation y-y = m ( x-x ). In our problem x = 0 ,y = 2 2 = 2 = 1, and m = 0 because its a horizontal line. Thus, y-1 = 0( x-1), which implies y = 1. (2 points)...
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