Chapter 9 Theory of Differential and Integral Calculus Section 1 Differentiability of a Function Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit provided this limit exists. If this limit exists for each x in an open interval I, then we say that f is differentiable on I.

E.g.1.2 (a) Show that f(x)=|tanx| is not differentiable at x=n*Pi

Proof of e.g.1.2

Discussion: Ex.9.1, Q.1, 6

Solution to Ex.9.1Q.1

Discussion Ex.9.1, Q.9

Section 2 Mean Value Theorem Rolle's Theorem. Let f be a function which is differentiable on the closed interval [a, b]. If f(a) = f(b) then there exists a point c in (a, b) such that f '(c) = 0.

Example of Rolle’s Theorem

Mean Value Theorem Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that

Proof of Mean Value Theorem

Example 1 of Mean Value Theorem

Example 2 of Mean Value Theorem

Corollaries (1) Let f be a differentiable function whose derivative is positive on the closed interval [a, b]. Then f is increasing on [a, b]. (2) Let f be a differentiable function whose derivative is negative on the closed interval [a, b]. Then f is decreasing on [a, b].

Proof of Corollary(1)

Proof of corollary(2)

First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. (1) If f '(x) > 0 for all x in (a, c) and f '(x) < 0 for all x in (c, b), then c is a local maximum of f. (2) If f '(x) < 0 for all x in (a, c) and f '(x) > 0 for all x in (c, b), then c is a local minimum of f.

Corollary 3 If f ’(x)=0 for all x in an interval I, then f(x) is a constant function in I.

Corollary 4 If f and g are differentiable functions on I and f ’(x)=g’(x) for all x in I, then f(x)=g(x) +c for some constant c.

Example:

Section 3 Convex Functions Definition 3.1 A real-valued function f(x) defined on an interval I is said to be convex on I iff for any two points x 1, x 2 in I and any two positive numbers p and q with p+q=1, f(px 1 +qx 2 )  pf(x 1 ) + qf(x 2 ).

Example

Theorem 3.2

Proof of Theorem 3.2

Example

§4 Definite Integral is the Limit of a Riemann Sum

n y=f(x)Find the sum of the areas of the rectangles in terms of n and f. y=f(x) AiAi

Group Discussion Express each of the following integrals as a limit of sum of areas:

Group Discussion Express each of the limits as a definite integral :

Example 4.1(a) Read example 4.1(b) Classwork Ex.9.4 Q.3 Read example 4.1(b) Classwork Ex.9.4 Q.3

Area bounded by the curve, x-axis, x=a and x=b Homework Ex.9.4

Example 4.1(a)

Section 5 Properties of Definite Integrals When does the equality fail? Theorem 5.4 Discussion:Ex.9.5, Q.1,2

When does the equality fail? Corollary 5.5 Discussion: Ex.9.5, Q.3

Corollary 5.3 |x|  K iff -K  x  K

f(b) f(a) ab

Example 5.1 Prove the following: Where do they come from?

Homework:Ex.9.5,3-7

Example 5.2

How to get n+1? How to get n?

What kinds of nos are they? How to get contradiction?

M m L

Section 6 Theorem 6.1 Mean Value Theorem of Integral A A ab

= L M m L

Differentiation of Integrals

Theorem 6.2 Fundamental Theorem of Calculus Since f(x) is continuous.

Newton-Leibniz Formula

Newton-Leibniz’s contribution in Calculus

What will happen if we don’t know Newton-Leibniz’s Theorem?

Questions for discussion

Application of Fundamental Theorem of Calculus

Example 2

Example 3

Example 4

Example 5

a f(a) u f(u) b

a f(a) u f(u) b 0