Differentiation - Quick Review From Calculus

Transcription

1 Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13

2 Introduction In this section, we quickly review the definition of the derivative. We also state and prove the basic differentiation theorems studied in a traditional calculus class. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 2 / 13

3 Definitions Throughout this section, unless stated otherwise, I will be an interval in R, f : I R is a function, and c I. We begin with the usual definition of the derivative. Definition (derivative) We recall the following definitions from differential calculus: 1 The derivative of f at c, denoted by f (c), is defined by f f (x) f (c) f (c + h) f (c) (c) = lim = lim x c x c h 0 h 2 If the limit exists, we say that f is differentiable at c. 3 f is differentiable on D I if f is differentiable at every point in D. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 3 / 13

4 One-Sided Derivatives Definition Similar to limits, we have one-sided derivatives. 1 The right derivative of f at c, denoted f + (c) is defined to be: f + f (x) f (c) f (c + h) f (c) (c) = lim = lim x c + x c h 0 + h 2 The left derivative of f at c, denoted f + (c) is defined to be: Remark f f (x) f (c) f (c + h) f (c) (c) = lim = lim x c x c h 0 h The reader should note the difference between f + (c) and f (c + ). The later is a limit notation, it means lim x c + f (x). Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 4 / 13

5 More on Derivatives If we use the definition of limits, saying that f is differentiable at c means that there exists a number we denote f (c) such that for every ɛ > 0, there exists δ > 0 such that if x I and 0 < x c < δ then f (x) f (c) f (c) x c < ɛ. The derivative corresponds to the slope of the tangent to the graph of y = f (x) at x = c. Being differentiable is a strong property in the sense that functions which are differentiable will have many properties which will make working with them easier. To begin with, we compare being differentiable with the property we just finished studying: continuity. Theorem If f is differentiable at c, then f is continuous at c. The converse of this theorem is not true. Consider f (x) = x. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 5 / 13

6 More on Derivatives Remark A function f will not be differentiable at a point c if one of the situations below happens: 1 The graph of f has a sharp corner at c. (in this case, the right and left derivatives may exist, but will be different). 2 The graph of f has a vertical tangent at c (the slope of a vertical line is not defined). 3 f is not continuous at c (By the above theorem, if f were differentiable at such points, it would have to be continuous). Remark If c Int (I ) then f (c) exists if and only if both f + (c) and f (c) exists and are equal. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 6 / 13

7 Higher Order Derivatives Let us first remark that D (f ) D (f ) (why?). Given f : I R, f is a new function whose domain will be a subset of I. One can find the derivative function, f (x) or dy df (x) or, using the dx dx definition above: f (x) = lim h 0 f (x + h) f (x) h Definition (higher order derivatives) The second derivative, denoted f (x) or d 2 y dx 2 f (x) = ( f (x) ) Similarly, we can define the derivative of order n, denoted f (n) (x) or d n y dx n, in terms of the derivative of order n 1 by: f (n) (x) = ( f (n 1) (x) ). by Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 7 / 13

8 Higher Order Derivatives You will notice the different notation. For the derivatives of order 1 to 3, we use f, f, f. For higher orders, we use f (4), f (5),... Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 8 / 13

9 Higher Order Derivatives You will notice the different notation. For the derivatives of order 1 to 3, we use f, f, f. For higher orders, we use f (4), f (5),... The derivatives of orders 1 and 2 are the most often used, their interpretation is fairly easy to grasp. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 8 / 13

10 Higher Order Derivatives You will notice the different notation. For the derivatives of order 1 to 3, we use f, f, f. For higher orders, we use f (4), f (5),... The derivatives of orders 1 and 2 are the most often used, their interpretation is fairly easy to grasp. f (x) represents how f changes with respect to x. It is also the slope of the tangent to the graph of y = f (x). Its sign provides information about whether f is increasing or decreasing. If f gives the position of an object, then f is the velocity of the object. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 8 / 13

11 Higher Order Derivatives You will notice the different notation. For the derivatives of order 1 to 3, we use f, f, f. For higher orders, we use f (4), f (5),... The derivatives of orders 1 and 2 are the most often used, their interpretation is fairly easy to grasp. f (x) represents how f changes with respect to x. It is also the slope of the tangent to the graph of y = f (x). Its sign provides information about whether f is increasing or decreasing. If f gives the position of an object, then f is the velocity of the object. f (x) is the rate of change of f (x). Its sign provides information about the concavity of f. If f gives the position of an object, then f is the acceleration of the object. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 8 / 13

12 Rules of Differentiation We list the usual rules of differentiation encountered in a differential calculus course. Theorem Let f, g : I R be two differentiable functions at a I, let C be a constant. Then, f ± g, Cf, fg, f g 1 (Cf ) (a) = Cf (a) 2 (f ± g) (a) = f (a) ± g (a) 3 (fg) (a) = f (a) g (a) + f (a) g (a) ( ) f 4 (a) = f (a) g (a) f (a) g (a) g (g (a)) 2 (if g (a) 0) are differentiable at a and: Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 9 / 13

13 Rules of Differentiation: Chain Rule Theorem (chain rule) Let I and J be intervals of R, with f : I J and g : J R. Suppose that a I and assume that f is differentiable at a, and g is differentiable at f (a). Then, the composite function g f is differentiable at a and (g f ) (a) = g (f (a)) f (a) or (g (f (a))) = g (f (a)) f (a) When the chain rule is written with Leibniz notation, it is easier to remember. If we let y = f (x) and u = g (f (x)) = g (y), then the chain rule becomes: du dx = du dy dy dx Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 10 / 13

14 Chain Rule: Proof Outline Put y = f (x), y 0 = f (a) and define h : J R by g (y) g (y 0 ) g h (y) = (y 0 ) if y y 0 y y 0 0 if y = y 0 Show h is continuous at y 0 that is lim y y 0 h (y) = h (y 0 ) Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 11 / 13

15 Chain Rule: Proof Outline Put y = f (x), y 0 = f (a) and define h : J R by g (y) g (y 0 ) g h (y) = (y 0 ) if y y 0 y y 0 0 if y = y 0 Show h is continuous at y 0 that is lim y y 0 h (y) = h (y 0 ) Show h f is continuous at a, use it to prove that lim x a (h f ) (x) = 0 Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 11 / 13

16 Chain Rule: Proof Outline Put y = f (x), y 0 = f (a) and define h : J R by g (y) g (y 0 ) g h (y) = (y 0 ) if y y 0 y y 0 0 if y = y 0 Show h is continuous at y 0 that is lim h (y) = h (y 0 ) y y 0 Show h f is continuous at a, use it to prove that lim (h f ) (x) = 0 x a [ ] g (f (x)) g (f (a)) f (x) f (a) Prove = [h (f (x)) + g (y 0 )] x a x a Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 11 / 13

17 Chain Rule: Proof Outline Put y = f (x), y 0 = f (a) and define h : J R by g (y) g (y 0 ) g h (y) = (y 0 ) if y y 0 y y 0 0 if y = y 0 Show h is continuous at y 0 that is lim h (y) = h (y 0 ) y y 0 Show h f is continuous at a, use it to prove that lim (h f ) (x) = 0 x a [ ] g (f (x)) g (f (a)) f (x) f (a) Prove = [h (f (x)) + g (y 0 )] x a x a Using the definition and what we derived above, compute (g f ) (a) Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 11 / 13

18 Rules of Differentiation: Chain Rule Example Let f (x) = x 2. Using the definition, find f (x). Example { x 3 if x 0 Let f (x) = x 2. Find where f is differentiable, compute its if x < 0 derivative. Where is f continuous? Example { x Let f (x) = 2 sin 1 if x 0 x. Find where f is differentiable, 0 if x = 0 compute its derivative. Where is f continuous? Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 12 / 13

19 Exercises See the problems at the end of Differentiation: A Quick Review From Differential Calculus. Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 13 / 13