ep3 - 92.131 1) Calculus 1 Differentiability and Continuity...

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92.131 Calculus 1 Differentiability and Continuity 1) a) Write down the limit definition of the derivative of a function ) ( x f . b) Using part (a) calculate ) ( x f , for 2 6 ) ( + = x x f . How does the result compare to what you already know about linear functions? 2) a) Write down the limit definition of the derivative of a function ) ( x f . b) Using part (a) calculate ) ( x f , for 3 4 2 ) ( 2 + = x x x f . c) Write down the power rule, and use it to check your answer in part(b). 3) a) Write down the limit definition of the derivative of a function ) ( x f . b) Using part (a) calculate ) ( x f , for 2 3 3 ) ( 2 + = x x x f . c) Write down the power rule, and use it to check your answer in part(b). 4) a) Using the limit definition of the derivative, calculate ) ( x f , for 3 1 ) ( 3 x x x f + = . b) Use the power rule to check your answer. 5) a) Using the limit definition of the derivative, calculate ) ( x f , for 2 1 ) ( 2 2 x x x f + = . b) Use the power rule to check your answer. 6) Consider the function 3 2 ) ( | | 2 ) ( π + = x x x f . a) Write down the limit definition for the derivative of a function. b) Using the definition of the derivative, determine whether or not the function ) ( x f differentiable at 0 = x . Justify your steps using limit laws or theorems. c) Is ) ( x f continuous at at 0 = x ? Explain your reasoning. 7) Consider the function 1 | | ) ( 2 ) ( + + = x x x f . a) Using the definition of the derivative, determine whether or not the function ) ( x f differentiable at 0 = x . b) Is ) ( x f continuous at at 0 = x ? Explain your reasoning. 8) Suppose that the function ) ( x f is defined by the rule > + + = 2 2 1 ) ( 3 2 x a x x x a x f , where a is a real constant. What must the value of a be in order for the function to be continuous at 2 = x ? Justify your reasoning. 9) Suppose that the function ) ( x f is defined by the rule + < + = 1 1 1 ) ( 2 x x b x a x x f . What relationship must a and b satisfy for the function to be continuous at 1 = x ? 10) Using the limit definition of the derivative, calculate ) ( x f , for 2 3 ) ( x x f = . Use the power rule to check your answer.
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92.131 Calculus 1 Differentiability and Continuity Solutions 1) a) h x f h x f x f h ) ( ) ( lim ) ( 0 + = b) [] [ ] h x h x h x f h x f x f h h 2 6 2 ) ( 6 lim ) ( ) ( lim ) ( 0 0 + + + = + = 6 6 lim 2 6 2 6 6 lim 0 0 = = + + = h h h x h x h h Since 2 6 ) ( + = x x f is a linear function its graph is a line whose constant slope is
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ep3 - 92.131 1) Calculus 1 Differentiability and Continuity...

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