MTH101 Final Term Handouts

45-Planning Production Levels: Linear Programming VU

TABLE OF CONTENTS :

Lesson 1 :Coordinates, Graphs, Lines

Lesson 2 :Absolute Value

Lesson 3 :Coordinate Planes and Graphs

Lesson 4 :Lines

Lesson 5 :Distance; Circles, Quadratic Equations

Lesson 6 :Functions and Limits

Lesson 7 :Operations on Functions

Lesson 8 :Graphing Functions

Lesson 9 :Limits (Intuitive Introduction)

Lesson 10:Limits (Computational Techniques)

Lesson 11: Limits (Rigorous Approach)

Lesson 12 :Continuity

Lesson 13 :Limits and Continuity of Trigonometric Functions

Lesson 14 :Tangent Lines, Rates of Change

Lesson 15 :The Derivative

Lesson 16 :Techniques of Differentiation

Lesson 17 :Derivatives of Trigonometric Function

Lesson 18 :The chain Rule

Lesson 19 :Implicit Differentiation

Lesson 20 :Derivative of Logarithmic and Exponential Functions

Lesson 21 :Applications of Differentiation

Lesson 22 :Relative Extrema

Lesson 23 :Maximum and Minimum Values of Functions

Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem

Lesson 25 :Integrations

Lesson 26 :Integration by Substitution

Lesson 27 :Sigma Notation

Lesson 28 :Area as Limit

Lesson 29 :Definite Integral

Lesson 30 :First Fundamental Theorem of Calculus

Lesson 31 :Evaluating Definite Integral by Subsitution

Lesson 32 :Second Fundamental Theorem of Calculus

Lesson 33 :Application of Definite Integral

Lesson 34 :Volume by slicing; Disks and Washers

Lesson 35 :Volume by Cylindrical Shells

Lesson 36 :Length of Plane Curves

Lesson 37 :Area of Surface of Revolution

Lesson 38:Work and Definite Integral

Lesson 39 :Improper Integral

Lesson 40 :L’Hopital’s Rule

Lesson 41 :Sequence

Lesson 42 :Infinite Series

Lesson 43 :Additional Convergence tests

Lesson 44 :Alternating Series; Conditional Convergence

Lesson 45 :Taylor and Maclaurin Series

Lecture # 20

Derivatives of Logarithmic and Exponential Functions and Inverse functions and their derivatives

Derivative of the logarithmic function log ( ) b f x x

ln( ) f x x

Logarithmic Differentiation

x f x a

Derivative of the logarithmic function

log ( ) b y f x x

Recall the Logarithm of a real number. Let x, b be real, and an unknown real number y. What should y be so that if you raise y to b, you get x.

That is y b x

This is denoted as log ( ) b y f x x

b is called the base. The equation above is read as “log base b of x is y .We want to find the derivative of the log function. Before we do that, here are some formulas we will need that follow from the properties of log functions. You can refer to them in section 7.1 of your text.

Theorem 7.1.1

Let's apply the derivative on both sides and use its definition to get

lim(1 2.71 x x e x e

log ( log ( ) [log ( )] lim

lim log

lim log

lim log (1 b b b h

b h

b h

b v x h x dy d x dx dx h

x h

h x

h

h x

v vx Last step comes from letting

and then 0 as h v v h x

lim log (1

1 limlog (1

Using Theorem 7.1.1 part (e)

1 log lim(1

Using the fact that log is continuous

1 log b v

v b v

v b v

b v x v

v x

v x

e x

Thus

[log ( )] log b b d x e x dx x There exists a “change of base” formula which Let's you convert your given log to a certain base to another. This is helpful because modern calculators have two log button, one with base 10, written as only log (x) and the other for the Natural log to the base e written as ln (x). To keep calculations easy, we can rewrite the above formula for the derivative as

[log ( )] ln( ) b d x x dx x b

If the base happens to be the number e, then we get from this formula

ln( ) d x x dx x So you see that the the simplest formula for the derivative comes from base b = e. That’s why ln (x) is a very important function in terms of Calculus

ln( ) f x x

We will generalize the derivative formula we just found for the log or ln function to include composition of functions where one is the ln function, and the other is some function u(x) > 0 and differentiable at x.

[log ( )] ln( )

[ln( )] b d du u dx u b dx

d du u dx u dx Using the chain rule here