Difference between revisions of "Math 22 The Derivative and the Slope of a Graph"

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<math>=\frac{x^2+2xh+h^2-1-x^2+1}{h}=\frac{2xh+h^2}{h}=\frac{h(2x+h)}{h}=2xh</math>
 
<math>=\frac{x^2+2xh+h^2-1-x^2+1}{h}=\frac{2xh+h^2}{h}=\frac{h(2x+h)}{h}=2xh</math>
  
'''2)''' <math>f(x)=4x-1</math>
+
'''2)''' <math>f(x)=4x+1</math>
 
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
 
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|Consider <math>\frac {f(x+h)-f(x)}{h}=\frac {4(x+h)-1 -(4x-1)}{h}=\frac {4x+4h-1+4x+1}{h}=\frac {4h}{h}=4</math>
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|Consider <math>\frac {f(x+h)-f(x)}{h}=\frac {4(x+h)+1 -(4x+1)}{h}=\frac {4x+4h+1+4x-1}{h}=\frac {4h}{h}=4</math>
 
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Latest revision as of 15:15, 5 October 2020

Slope of a Graph

We can estimate the slope at the given point to be


Slope =

Difference Quotient

 The slope  of the graph of  at the point  can be 
 written as :
 
 
 
 The right side of this equation  is called Difference Quotient

Example: Find the Different Quotient of

1)

Solution: Consider

2)

Solution:  
Consider

Definition of the Derivattive

 The derivative of  at  is given by
 
 
 
 provided this limit exists. A function is differentiable at  when its 
 derivative exists at . The process of finding derivatives is called differentiation.

Example: Use limit definition to find the derivative of

1)

Solution: Consider:

2)

Solution:  
Consider:

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