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. 2 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017 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. 3 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017 Ex: Show that f(x) has no derivative at 2. Continuous at 2? One sided (derivative) limits at 2? 4 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017 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. 6 2.2 Day 2 The Derivative as a Function.notebook October 04, 2017 Homework: Posted on site 7.