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Unformatted text preview: Math 31A 2010.01.19 MATH 31A DISCUSSION JED YANG 1. Derivatives 1.1. Basics. Given a function f ( x ). The slope of the tangent line at x = c is f ( c ). 1.1.1. Power Rule. For all exponents n R , d dx x n = nx n 1 . Not for e x , x x . 1.1.2. Linearity Rules. If f and g are differentiable functions, c R , then cf and f + g are differentiable. Indeed, ( f + g ) = f + g and ( cf ) = cf . 1.1.3. Product and Quotient Rules. If f and g are differentiable, ( fg ) = fg + gf . And ( f/g ) = ( f g g f ) /g 2 . 1.2. Exercise 3.2.46. Sketch the graphs of f ( x ) = x 2 5 x +4 and g ( x ) = 2 x +3. Find the value of x at which the graphs have parallel tangent lines. Solution. We need f ( x ) = g ( x ). Notice f ( x ) = 2 x 5 and g ( x ) = 2. So we solve 2 x 5 = 2 to get x = 3 2 . square 1.3. Exercise 3.2.52. Show that if the tangent lines to the graph of y = 1 3 x 3 x 2 at x = a and x = b are parallel, then either...
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