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soln-prac-mid2

# soln-prac-mid2 - Solutions to Practic Midterm 2 1 Find the...

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Solutions to Practic Midterm 2 1 Find the normal line to the curve x + y = e 2 y at the point (1 , 0) . Solution We need to find dy dx so that we can find the slope of the normal line. Using implicit differentiation, we find 1 2 x + y (1 + y 0 ) = 2 y 0 e 2 y . We plug in the point (1 , 0) and solve for y 0 . Doing so, we get y 0 = 1 3 . Thus the normal line has slope - 3, and its equation is y = - 3( x - 1) . (Note, you could have also done implicit diff to x + y = e 4 y yielding 1 + dy dx = 4 ye 4 y dy dx and then plugging in y = 0 yields 1 + dy dx = 4 dy dx .) 2 Find ds dw if t = Tan - 1 (ln w ) and s = 2 t ( t + t ) 1 / 3 . Solution By the Chain Rule, ds dw = ds dt · dt dw . ds dt = 2( t + t ) 1 / 3 - 2 t · 1 3 ( t + t ) - 2 / 3 · 1 + 1 2 t ( t + t ) 2 / 3 dt dw = 1 1 + (ln w ) 2 · 1 w · 1 2 w Thus ds dw = 2( t + t ) 1 / 3 - 2 t · 1 3 ( t + t ) - 2 / 3 · 1 + 1 2 t ( t + t ) 2 / 3 · 1 1 + (ln w ) 2 · 1 w · 1 2 w 3 Find the derivative with respect to the appropriate independent variable.

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soln-prac-mid2 - Solutions to Practic Midterm 2 1 Find the...

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