2.2 Day 2 The Derivative as a Function.notebook October 04, 2017

Calculus ­ 2.2 The Derivative as a Function

Day 2: • Identify where a function has no derivative • Explore continuity with regards to derivatives • Work out homework examples

1 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017

Identifying where a function has no derivative: Since the derivative of a function involves a limit, it should make sense that a function f has no derivative at a point c if the limit below does not exist.

We will talk about 3 ways this can happen: 1. If the left hand limit at c does not equal the right hand limit at c. This creates a corner 2. If the one sided limits are both infinite and are both ∞ or both ­ ∞. This creates a vertical tangent line. 3. If the one sided limits are both infinite but one equals ­∞ and the other equals ∞. This creates a cusp (also creating a vertical tangent line).

Functions are also only differentiable at x values where f(x) is continuous.

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Ex: Does f ' (1) exist given f(x)?

First: Make sure that f(x) is continuous at 1, if it's not continuous, it's not differentiable.

Second: Check the one sided limits. (not limits of f(x), but limits that would create f '(x))

Third: Interpret and state whether you have a corner or tangent line.

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Ex: Show that f(x) has no derivative at 2.

Continuous at 2?

One sided (derivative) limits at 2?

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Ex: Given the graph of f ' (x) , answer the following questions.

a) does the graph of f have any horizontal tangent lines? If yes, explain why and where it occurs.

Remember that f '(x) models the slopes at the x­value and plots them as y­values. Then think about what the slope of f would be to create horizontal tangent lines. That can only happen if the slope is 0. The y value is 0 at ­2 and 4. so to answer the question, the graph of f has horizontal tangent lines at (­2, f(­2)) and (4, f(4))

b) Does the graph of f have any vertical tangent lines? If so, state where and why they occur.

Vertical tangent lines happen at a point where the derivative (at that point) is ± ∞ coming from the left/right. Since f '(x) is headed towards positive infinity from both sides at x = 0, f has a vertical tangent line at (0, f(0))

c) Does the graph of f have any corners? If yes, explain why and identify where it happens.

Corners exist where the derivative has a left hand limit Not equaling the right hand limit. This happens at x = 2 on f '(x). So a corner exists at (2, f(2)).

5 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017

Differentiability:

Ex: Determine if the Heaviside function has a derivative at 0.

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Homework: Posted on site

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