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Miss Battaglia BC Calculus

2.1 Differentiability Objective: Understand the relationship between differentiability and continuity. Miss Battaglia BC Calculus. Differentiability & Continuity. Alternative limit form of the derivative:

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Miss Battaglia BC Calculus

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  1. 2.1 DifferentiabilityObjective: Understand the relationship between differentiability and continuity Miss Battaglia BC Calculus

  2. Differentiability & Continuity Alternative limit form of the derivative: provided this limit exists. Note the limit in this alternative form requires that the one-sided limits and exist and are equal. These one-sided limits are called the derivatives from the left and from the right. It follows that f is differentiable on the closed interval [a,b] if it is differentiable on (a,b) and if the derivative from the right at a and the derivative from the left at b both exist.

  3. A Graph with a Sharp Turn • Use the alternative form of the derivative to find the derivative of f(x)=|x-2| at 2.

  4. A Graph with a Vertical Tangent Line Use the alternative form of the derivative to find the derivative of f(x)=x1/3 at 0.

  5. Example • Use the alternative form of the derivative to find the derivative of g(x)=x(x-1) at 1.

  6. Example • Use a graphing calculator to graph the function and find the x-values at which f is differentiable. f(x)=|x-5|

  7. Thm 2.1 Differentiability Implies Continuity If f is differentiable at x=c then f is continuous at x=c. You can prove f is continuous at x=c by showing f(x) approaches f(c) as xc. Proof:

  8. Classwork/Homework • Page 104 #59, 60, 75-81 odd, 83-92, 101-104

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