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samplequiz1 - SOLUTIONS TO QUIZ 1 Question 1[6marks Let f x...

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Unformatted text preview: SOLUTIONS TO QUIZ 1 Question 1 [6marks] Let f ( x ) be the function defined by f ( x ) = q 1 + 1 /x , x > 0. 1(a) Find the derivative f ( x ) using only first principles. 1(b) Find an equation of the tangent line to the graph of y = f ( x ) at x = 1 / 3 . Solution to Question 1: 1(a) f ( x ) = lim h → f ( x + h )- f ( x ) h = lim h → q 1 + 1 / ( x + h )- q 1 + 1 /x h = lim h → 1 + 1 / ( x + h )- (1 + 1 /x ) h ( q 1 + 1 / ( x + h ) + q 1 + 1 /x ) = lim h →- 1 x ( x + h )( q 1 + 1 / ( x + h ) + q 1 + 1 /x ) =- 1 2 x 2 q 1 + 1 /x . 1(b) By a simple computation f (1 / 3) = 2 and f (1 / 3) =- 9 / 4, and therefore the tangent line is given by y- 2 =- 9 / 4( x- 1 / 3). Question 2 [6 marks] Compute the following limits. 2(a) lim x →∞ ( √ x 2 + ax- √ x 2- ax ), where a is a constant. 2(b) lim x → 1 ( x 2- 1) / ( √ x + 8- 3). 2(c) lim x → (2 x + 3 x 2 ) / (3 x- 2 x 2 ). Solution to Question 2: 2(a) lim x →∞ ( √ x 2 + ax- √ x 2- ax ) = lim x →∞ x 2 + ax- ( x 2- ax...
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samplequiz1 - SOLUTIONS TO QUIZ 1 Question 1[6marks Let f x...

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