samplemt4 - SOLUTIONS TO MIDTERM #1 QUESTION 1: [10 marks]...

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SOLUTIONS TO MIDTERM #1 QUESTION 1: [10 marks] (a) Give the definition of the derivative f 0 ( x )o fafunct ion f ( x ). (b) What is the equation of the tangent line to the graph of y = f ( x )at x = a ? (c) Using only the definition of the derivative find f 0 ( x ) for the function f ( x )= 1 x . Solutions: (a) f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h . (b) y - f ( a f 0 ( a )( x - a ) . (c) The derivative of f ( x x is calculated as follows: f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 1 / x + h - 1 / x h = lim h 0 x - x + h h x x + h = lim h 0 ± x - x + h h x x + h × x + x + h x + x + h ² = lim h 0 x - ( x + h ) h x x + h ( x + x + h ) = lim h 0 - 1 x x + h ( x + x + h ) = - 1 2 x x by putting h = 0 and using continuity. QUESTION 2: [12 marks] (a) Find the derivative of f ( x p g ( x x = - 1i f g ( - 1) = 4 and g 0 ( - 1) = - 2. (b) Find the derivative of f ( x x - 1 g ( x )+1 at x =0if g (0)=2and g 0 (0) = 2. (c) Find the derivative of f ( x )=s in ( πg ( x )) at x = a if you are given that g ( a 2 3 and g 0 ( a b .
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samplemt4 - SOLUTIONS TO MIDTERM #1 QUESTION 1: [10 marks]...

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