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Aim: How do we determine if a function is differential at a point?

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Presentation on theme: "Aim: How do we determine if a function is differential at a point?"— Presentation transcript:

1 Aim: How do we determine if a function is differential at a point?
Do Now: Find the instantaneous rate of change of a given function at (2,4)

2 Definition of Derivative
Mathboat.com

3 Prove that left-hand and right-hand derivatives are not equal
exists if_________ Example 1 Prove that left-hand and right-hand derivatives are not equal Mathboat.com

4 Example 2 Check if left-hand and right-hand derivatives are equal at x=0. Mathboat.com

5 Applications of Derivative
Notations of Derivative Why quotient ? Next slide… Not common Derivatives of the higher order Mathboat.com

6 Explanation of the different notation?

7 Example 3 Mathboat.com

8 A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

9 Differentiability on a Closed Interval
Mathboat.com

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11

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13 Example 1 Determine whether the following derivatives exist for the graph of the function.

14 The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.

15 The Slope of the Graph of a Line
Example: Find the slope of the graph of at the point (2, 5).

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17 Example 4 vertical tangent Mathboat.com

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19 Mathboat.com

20 vertical tangent vertical tangent vertical tangent y Mathboat.com

21 Example 5 -5 Mathboat.com

22 The following is a correct statement (left side = right side):
Theorem Proof: The following is a correct statement (left side = right side): Mathboat.com

23

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