samplemt3 - SOLUTIONS TO MIDTERM #1: MATH 102, SECTIONS 102...

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SOLUTIONS TO MIDTERM #1: MATH 102, SECTIONS 102 & 105 QUESTION 1: [6 marks] (a) Give the definition of the derivative f 0 ( x )o fafunct ion f ( x ) . (b) In a few brief sentences give several interpretations of the derivative of a function f ( x ) . (c) State the formula for the tangent line approximation. (a) The derivative is defined to be f 0 ( x 0 ) = lim h 0 f ( x 0 + h ) - f ( x 0 ) h . (b) The derivative f 0 ( x 0 ) can be interpreted as the slope of the graph y = f ( x )when x = x 0 . It is also the instantaneous rate of change of y = f ( x ) with respect to x. If y = f ( t )represents distance travelled than y 0 ( t )= v ( t ) represents velocity and v 0 ( t y 00 ( t a ( t )rep re s en t s acceleration. (c) The tangent line approximation is f ( a + h ) f ( a )+ hf 0 ( a ) . Equivalently the tangent line approximation is f ( a + h f ( a hf 0 ( a E ( h ) , where E ( h ) represents the error made in the approximation f ( a + h ) f ( a hf 0 ( a ) . y=f(x) a a+h E(h) f(a)+hf’(a) y=f(a)+(x-a)f’(a) 1
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2 QUESTION 2: [6 marks] (a) Using only the definition of the derivative, and not the rules of differentiation, find the derivative of f ( x )=1 / x. (b) Find the equation of the tangent line of y =1 / x when x =25 . (c) Using the tangent line approximation estimate the value of 1 / 26 .
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samplemt3 - SOLUTIONS TO MIDTERM #1: MATH 102, SECTIONS 102...

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