Objective Understand the relationship between differentiability and continuity Miss Battaglia BC Calculus Differentiability amp Continuity Alternative limit form of the derivative provided this limit exists Note the limit in this alternative form requires that the onesided limits ID: 264689
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Slide1
2.1 DifferentiabilityObjective: Understand the relationship between differentiability and continuity
Miss
Battaglia
BC CalculusSlide2
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.Slide3
Use the alternative form of the derivative to find the derivative of f(x)=|x-2| at 2.
A Graph with a Sharp TurnSlide4
A Graph with a Vertical Tangent Line
Use the alternative form of the derivative to find the derivative of f(x)=x
1/3
at 0.Slide5
Use the alternative form of the derivative to find the derivative of g(x)=x(x-1) at 1.
ExampleSlide6
Use a graphing calculator to graph the function and find the x-values at which f is differentiable.
f(x)=|x-5|
ExampleSlide7
Thm
2.1 Differentiability Implies Continuity
If f is differentiable at x=c then f is continuous at x=c.
Proof:You can prove f is continuous at x=c by showing f(x) approaches f(c) as xc
.Slide8
Page 104 #59, 60, 75-81 odd, 83-92, 101-104
Classwork/Homework