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Differentiability and Piecewise Functions
What are the three things that make a function not differentiable ?Not continuous at the point Vertical tangent line at the point A cusp at the point.
Where is the derivative undefined?1. …at a point of discontinuity. Example: f(x) is not differentiable at x = ____
Where is the derivative undefined?2. …where there is a vertical tangent line. Example: y = x1/3 is not differentiable at x = ___
3. …if the graph has a sharp point (cusp)Where is the derivative undefined? 3. …if the graph has a sharp point (cusp) f(x) is not differentiable at x = ____ f(x) is not differentiable at x = ____
Determine the value(s) of x at which the function is not differentiable. Give the reason.
Derivatives of piecewise functions…If then, f ’(x)= f ’(5)= f ’(-2)=
Is the function continuous? Justify your answer.Is the function differentiable at x = 1? Justify your answer.
Determine whether the function is differentiable at x = 3Determine whether the function is differentiable at x = 3. Justify your answer.
Derivatives of Absolute value functions…If then, f ’(x)= f ’(5)= f ’(-2)=
Is the function differentiable everywhere?
DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,
I’m going nuts over derivatives!!! 2.1 The Derivative and the Tangent Line Problem.
2.1 Tangent Line Problem. Tangent Line Problem The tangent line can be found by finding the slope of the secant line through the point of tangency and.
2.1 The derivative and the tangent line problem
The Derivative As A Function. 2 The First Derivative of f Interpretations f ′(a) is the value of the first derivative of f at x = a. f ′(x) is.
Derivative as a Function
Limit Definition of the Derivative. Objective To use the limit definition to find the derivative of a function. TS: Devoloping a capacity for working.
Section 3.2b. The “Do Now” Find all values of x for which the given function is differentiable. This function is differentiable except possibly where.
Assigned work: pg 74#5cd,6ad,7b,8-11,14,15,19,20 Slope of a tangent to any x value for the curve f(x) is: This is know as the “Derivative by First Principles”
The Derivative. Definition Example (1) Find the derivative of f(x) = 4 at any point x.
Limit & Derivative Problems Problem…Answer and Work…
Chapter 2 Section 2 The Derivative!. Definition The derivative of a function f(x) at x = a is defined as f’(a) = lim f(a+h) – f(a) h->0 h Given that a.
2.1 The Derivative and the Tangent Line Problem
SECTION 3.1 The Derivative and the Tangent Line Problem.
Section Continuity. continuous pt. discontinuity at x = 0 inf. discontinuity at x = 1 pt. discontinuity at x = 3 inf. discontinuity at x = -3 continuous.
Thoughts The function F(x) has the following graph. Match the following Values. 1)F’(-3) = 2)F’(-2) = 3)F’(0) = (a) 1 (b) -2 (c) 0.
Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.
3.3 Rules for Differentiation AKA “Shortcuts”. Review from places derivatives do not exist: ▫Corner ▫Cusp ▫Vertical tangent (where derivative is.
3.2 & 3.3. State the Differentiability Theorem Answer: If a function is differentiable at x=a, then the function is continuous at x=a.
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