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Some of 2.3, and all of 2.1 Rates of Change, Slopes, and Derivatives ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
A. What algebra skill you’ll need for 8.2 ,[object Object],[object Object],[object Object]
B. Where average rate of change (ARC) is on a graph ,[object Object],[object Object],[object Object]
C. How to find it algebraically (without a graph) ,[object Object],[object Object],[object Object]
 
 
D. How we get the difference quotients definition of ARC ,[object Object],[object Object]
Let’s take this curve, and label a fixed point x. It would have corresponding y-value labeled f(x). I am going to jog over a certain distance  h  on the x-axis. What could I call this new fixed point?
x + h  would be that newly created point  on the x-axis. Its corresponding  y-value would be called  f(x + h).
Here is the red, dotted secant line.  We need the slope of it.
RISE = RUN =
E. Where instantaneous rate of change (IRC) is on a graph ,[object Object]
Sometimes they ask you whether the slope of the tangent line is positive, negative, or zero.
F. How we get the difference quotient definition of IRC Remember the  secant  line that was  h  wide?   ,[object Object]
That would get closer and closer to the slope of the tangent line! The IRC! This will be useful because we can use this formula  algebraically  (no graphing necessary to find or guess at the IRC.)  Let’s see if we can work one…
G. How to find the IRC ,[object Object],STATE FORMULA PLUG IN X = 1 f(1 + h) would be _________ _______________________ f(1) would be _________ SIMPLIFY CANCEL THE h’S SUB IN ZERO FOR h
 
 
 
H. What a derivative is ,[object Object],[object Object],[object Object]
 
 
I. What Leibniz’s Notation looks like
J. Word Problems ,[object Object],[object Object],[object Object]
 
 
Use the limit definition to find an EQUATION (y=mx+b) of the tangent line to the graph of f at the given point:
The word “differentiable” means… _________________________________ ,[object Object],[object Object],[object Object],[object Object]
What suggested HW problems to try? ,[object Object],[object Object]

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125 arc, irc, and derivative

  • 1.
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  • 4.
  • 5.  
  • 6.  
  • 7.
  • 8. Let’s take this curve, and label a fixed point x. It would have corresponding y-value labeled f(x). I am going to jog over a certain distance h on the x-axis. What could I call this new fixed point?
  • 9. x + h would be that newly created point on the x-axis. Its corresponding y-value would be called f(x + h).
  • 10. Here is the red, dotted secant line. We need the slope of it.
  • 12.
  • 13. Sometimes they ask you whether the slope of the tangent line is positive, negative, or zero.
  • 14.
  • 15. That would get closer and closer to the slope of the tangent line! The IRC! This will be useful because we can use this formula algebraically (no graphing necessary to find or guess at the IRC.) Let’s see if we can work one…
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  • 17.  
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  • 19.  
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  • 21.  
  • 22.  
  • 23. I. What Leibniz’s Notation looks like
  • 24.
  • 25.  
  • 26.  
  • 27. Use the limit definition to find an EQUATION (y=mx+b) of the tangent line to the graph of f at the given point:
  • 28.
  • 29.