Download presentation

Presentation is loading. Please wait.

Published byBritney Bowker Modified over 2 years ago

1
**2.1 The derivative and the tangent line problem**

2
**Definition of the derivative of a function**

The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a function of x.

3
**Notation for derivatives:**

f’(x) “f prime of x” dy “the derivative of y with respect to x” dx “ dy – dx” y’ “y prime” d [f(x)] dx Dx[y]

4
dy = lim ∆y dx ∆x ∆x = lim f(x + ∆x) – f(x) ∆x ∆x = f’(x)

5
**Alternative Form of a Derivative**

f’(c) = lim f(x) - f(c) x c x - c

6
**f’(c) = lim f(x) - f(c) x c- x - c f’(c) = lim f(x) - f(c) x c+ x - c**

The existence of the limit requires that the one-sided limits exist and are equal. f’(c) = lim f(x) - f(c) x c x - c (The derivative from the left) f’(c) = lim f(x) - f(c) x c x - c (The derivative from the right)

7
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.

8
Example: f(x) = |x – 2|

9
**Conditions where a function is not differentiable:**

1. At a point at which a graph has a sharp turn. 2. At a point at which a graph has a vertical tangent line. 3. At a point at which the function is not continuous.

10
**Theorem: Differentiability Implies Continuity**

If “f” is differentiable at x = c, then f is continuous at x = c.

Similar presentations

Presentation is loading. Please wait....

OK

2.1 The Derivative and the Tangent Line Problem.

2.1 The Derivative and the Tangent Line Problem.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on e-mail spam Ppt on networking related topics to economics Ppt on power system restructuring Ppt on organic farming in india Ppt on dry cell and wet cellulose Ppt on paintings and photographs related to colonial period of american Ppt on layer 3 switching Ppt on fmcg industry in india Ppt on advertising media planning Ppt on barack obama leadership quotes