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Limits and Derivatives
Concept of a Function
y  is a function of  x , and the relation  y  =  x 2  describes a function. We notice that with such a relation,  every value of  x  corresponds to one (and only one) value of  y . y  =  x 2
Since the value of  y  depends on a given value of  x , we call  y  the  dependent variable  and  x  the  independent variable  and of the function  y  =  x 2 .
 
 
 
Notation for a Function :  f ( x )
 
 
 
 
 
 
 
 
 
 
The Idea of Limits
Consider the function   The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x )
Consider the function   The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x ) 3.9 3.99 3.999 3.9999 un-defined 4.0001 4.001 4.01 4.1
Consider the function   The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 g ( x ) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 x y O 2
If a function  f ( x ) is a continuous at x 0 , then   .  approaches to, but not equal to
Consider the function   The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 g ( x )
Consider the function   The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 h ( x ) -1 -1 -1 -1 un-defined 1 2 3 4
does not exist.
A function  f ( x ) has limit  l  at  x 0  if  f ( x ) can be made as close to  l  as we please by taking  x  sufficiently close to (but not equal to)  x 0 . We write
Theorems On Limits
Theorems On Limits
Theorems On Limits
Theorems On Limits
Limits at Infinity
Limits at Infinity Consider
Generalized, if then
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorem where θ is measured in  radians . All angles in  calculus  are measured in  radians.
The Slope of the Tangent to a Curve
The Slope of the Tangent to a Curve The slope of the tangent to a curve  y  =  f ( x ) with respect to  x  is defined as provided that the limit exists.
Increments The increment △ x  of a variable is the change in  x  from a fixed value  x  =  x 0  to another value  x  =  x 1 .
For any function  y  =  f ( x ), if the variable  x  is given an increment △ x  from  x  =  x 0 , then the value of  y  would change to  f ( x 0  + △ x ) accordingly. Hence thee is a corresponding increment of  y (△ y ) such that  △ y  =  f ( x 0  + △ x ) –  f ( x 0 ) .
Derivatives (A) Definition of Derivative. The derivative of a function  y  =  f ( x ) with respect to  x  is defined as provided that the limit exists.
The derivative of a function  y  =  f ( x ) with respect to x is usually denoted by
The process of finding the derivative of a function is called  differentiation . A function  y  =  f ( x ) is said to be  differentiable  with respect to  x  at  x  =  x 0  if the derivative of the function with respect to  x   exists  at  x  =  x 0 .
The value of the derivative of  y  =  f ( x ) with respect to  x  at  x  =  x 0  is denoted  by  or  .
To obtain the derivative of a function by its definition is called  differentiation  of the function  from first principles .
[object Object],DERIVATIVES OF TRIGONOMETRIC FUNCTIONS Figure 3.4.1, p. 149
[object Object],DERIVS. OF TRIG. FUNCTIONS Equation 1
[object Object],DERIVS. OF TRIG. FUNCTIONS
[object Object],DERIVS. OF TRIG. FUNCTIONS
[object Object],[object Object],DERIVS. OF TRIG. FUNCTIONS Equation 2
[object Object],DERIVS. OF TRIG. FUNCTIONS
DERIVS. OF TRIG. FUNCTIONS Equation 3
[object Object],[object Object],DERIVS. OF TRIG. FUNCTIONS Formula 4
[object Object],[object Object],Example 1 DERIVS. OF TRIG. FUNCTIONS Figure 3.4.3, p. 151
[object Object],Formula 5 DERIV. OF COSINE FUNCTION
DERIV. OF TANGENT FUNCTION Formula 6
[object Object],[object Object],DERIVS. OF TRIG. FUNCTIONS
[object Object],[object Object],Example 2 DERIVS. OF TRIG. FUNCTIONS
[object Object],Example 2 Solution: tan2  x  + 1 =  sec2  x
[object Object],[object Object],Example 4 DERIVS. OF TRIG. FUNCTIONS
[object Object],[object Object],[object Object],Example 4 Solution:
[object Object],[object Object],Example 5 DERIVS. OF TRIG. FUNCTIONS
[object Object],Example 5 Solution:
[object Object],[object Object],Example 6 DERIVS. OF TRIG. FUNCTIONS
[object Object]

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Limits And Derivative slayerix