Lesson 6: The derivative as a function

Slide 1: Section 2.8 The Derivative as a Function Math 1a February 13, 2008 Announcements Office Hours TW 2–4 in SC 323 ALEKS is due Wednesday 2/20 HW on website

Slide 2: Outline Cleanup: Derivatives of some root functions The derivative function Worksheet #1 How can a function fail to be differentiable? Other notations The second derivative Worksheet #2

Slide 3: Last time: Worksheet problems 3 and 4 Problem Let f (x) = x 1/3 . Find f (x) and its domain. Problem Let f (x) = x 2/3 . Find f (x) and its domain.

Slide 4: Last time: Worksheet problems 3 and 4 Problem Let f (x) = x 1/3 . Find f (x) and its domain. Answer 1 f (x) = x −2/3 . The domain is all numbers except 0. 3 Problem Let f (x) = x 2/3 . Find f (x) and its domain.

Slide 5: Last time: Worksheet problems 3 and 4 Problem Let f (x) = x 1/3 . Find f (x) and its domain. Answer 1 f (x) = x −2/3 . The domain is all numbers except 0. 3 Problem Let f (x) = x 2/3 . Find f (x) and its domain. Answer 2 f (x) = x −1/3 . The domain is all numbers except 0. 3

Slide 6: Outline Cleanup: Derivatives of some root functions The derivative function Worksheet #1 How can a function fail to be differentiable? Other notations The second derivative Worksheet #2

Slide 7: The derivative function We have snuck this in: If f is a function, we can compute the derivative f (x) at each point x where f is differentiable, and come up with another function, the derivative function. What can we say about this function f ?

Slide 8: Worksheet #1

Slide 9: Outline Cleanup: Derivatives of some root functions The derivative function Worksheet #1 How can a function fail to be differentiable? Other notations The second derivative Worksheet #2

Slide 10: Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a.

Slide 11: Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have f (x) − f (a) lim (f (x) − f (a)) = lim · (x − a) x→a x→a x −a f (x) − f (a) = lim · lim (x − a) x→a x −a x→a = f (a) · 0 = 0

Slide 12: Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have f (x) − f (a) lim (f (x) − f (a)) = lim · (x − a) x→a x→a x −a f (x) − f (a) = lim · lim (x − a) x→a x −a x→a = f (a) · 0 = 0 Note the proper use of the limit law: if the factors each have a limit at a, the limit of the product is the product of the limits.

Slide 13: How can a function fail to be differentiable? Kinks f (x) x

Slide 14: How can a function fail to be differentiable? Kinks f (x) f (x) x x

Slide 15: How can a function fail to be differentiable? Cusps f (x) x

Slide 16: How can a function fail to be differentiable? Cusps f (x) f (x) x x

Slide 17: How can a function fail to be differentiable? Vertical Tangents f (x) x

Slide 18: How can a function fail to be differentiable? Vertical Tangents f (x) f (x) x x

Slide 19: How can a function fail to be differentiable? Weird, Wild, Stuff f (x) x

Slide 20: How can a function fail to be differentiable? Weird, Wild, Stuff f (x) f (x) x x

Slide 21: Outline Cleanup: Derivatives of some root functions The derivative function Worksheet #1 How can a function fail to be differentiable? Other notations The second derivative Worksheet #2

Slide 22: Notation Newtonian notation f (x) y (x) y Leibnizian notation dy d df f (x) dx dx dx

Slide 23: Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687

Slide 24: Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute

Slide 25: Outline Cleanup: Derivatives of some root functions The derivative function Worksheet #1 How can a function fail to be differentiable? Other notations The second derivative Worksheet #2

Slide 26: The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change!

Slide 27: The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change! Leibnizian notation: d 2y d2 d 2f f (x) dx 2 dx 2 dx 2

Slide 28: Worksheet #2