Some of 2.3, and all of 2.1 Rates of Change, Slopes, and Derivatives <ul><li>What algebra skill you’ll need for 8.2 </li><...
A. What algebra skill you’ll need for 8.2 <ul><li>Negative, positive slope </li></ul><ul><li>Slope formula </li></ul><ul><...
B. Where average rate of change (ARC) is on a graph <ul><li>The average rate of change(ARC) between  x =  2 and  x =  5 is...
C. How to find it algebraically (without a graph) <ul><li>Well, since the ARC is really a SLOPE, let’s recall the slope fo...
 
 
D. How we get the difference quotients definition of ARC <ul><li>You might have been introduced to the famous “difference ...
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...
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 <ul><li>It’s the slope of the  tangent  line. I’ll draw it: </li...
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?   <ul><li>It was  h ...
That would get closer and closer to the slope of the tangent line! The IRC! This will be useful because we can use this fo...
G. How to find the IRC <ul><li>Find the instantaneous rate of change of this function at x = 1. </li></ul>STATE FORMULA PL...
 
 
 
H. What a derivative is <ul><li>We can evaluate the IRC at any point we want algebraically, but wouldn’t it be easier if w...
 
 
I. What Leibniz’s Notation looks like
J. Word Problems <ul><li>First of all, the units of the ARC or IRC is always consistent with it being a  rate . [Like, “so...
 
 
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… _________________________________ <ul><li>If it is differentiable at a point, then it is ...
What suggested HW problems to try? <ul><li>In section 2.3, try numbers 3-11 odd, 13a, 17a. We will do more of section 2.3 ...
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125 arc, irc, and derivative

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125 arc, irc, and derivative

  1. 1. Some of 2.3, and all of 2.1 Rates of Change, Slopes, and Derivatives <ul><li>What algebra skill you’ll need for 8.2 </li></ul><ul><li>Where average rate of change (ARC) is on a graph </li></ul><ul><li>How to find it algebraically (without a graph) </li></ul><ul><li>How we get the difference quotients definition of ARC </li></ul><ul><li>Where instantaneous rate of change (IRC) is on a graph </li></ul><ul><li>How we get the difference quotient definition of IRC </li></ul><ul><li>How to find the IRC </li></ul><ul><li>What a derivative is </li></ul><ul><li>What Leibniz’s Notation looks like </li></ul><ul><li>Word Problems </li></ul>
  2. 2. A. What algebra skill you’ll need for 8.2 <ul><li>Negative, positive slope </li></ul><ul><li>Slope formula </li></ul><ul><li>Combining rational expressions </li></ul>
  3. 3. B. Where average rate of change (ARC) is on a graph <ul><li>The average rate of change(ARC) between x = 2 and x = 5 is the SLOPE of that red dotted line. </li></ul><ul><li>[a secant line] </li></ul><ul><li>Let’s find it: </li></ul>
  4. 4. C. How to find it algebraically (without a graph) <ul><li>Well, since the ARC is really a SLOPE, let’s recall the slope formula: </li></ul><ul><li>We can use that f(x) notation because we’ll be dealing with functions. Recall that f(x) simply means the y value that corresponds to x. </li></ul><ul><li>Let’s see if we can apply this to a problem: </li></ul>
  5. 7. D. How we get the difference quotients definition of ARC <ul><li>You might have been introduced to the famous “difference quotient” in algebra. In case you weren’t, here is what it looks like: </li></ul><ul><li>I’ll show you where it came from. In part F, you’ll find out what is so great about it. </li></ul>
  6. 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?
  7. 9. x + h would be that newly created point on the x-axis. Its corresponding y-value would be called f(x + h).
  8. 10. Here is the red, dotted secant line. We need the slope of it.
  9. 11. RISE = RUN =
  10. 12. E. Where instantaneous rate of change (IRC) is on a graph <ul><li>It’s the slope of the tangent line. I’ll draw it: </li></ul>
  11. 13. Sometimes they ask you whether the slope of the tangent line is positive, negative, or zero.
  12. 14. F. How we get the difference quotient definition of IRC Remember the secant line that was h wide? <ul><li>It was h wide. Imagine you could make h shrink (“approach zero”). </li></ul>
  13. 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…
  14. 16. G. How to find the IRC <ul><li>Find the instantaneous rate of change of this function at x = 1. </li></ul>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
  15. 20. H. What a derivative is <ul><li>We can evaluate the IRC at any point we want algebraically, but wouldn’t it be easier if we could create a function for the IRC, and then we could plug in any value. It would be more efficient. </li></ul><ul><li>The DERIVATIVE is just that. Basically, it is a function for the IRC. </li></ul><ul><li>When you see “IRC,” think “Derivative.” </li></ul>
  16. 23. I. What Leibniz’s Notation looks like
  17. 24. J. Word Problems <ul><li>First of all, the units of the ARC or IRC is always consistent with it being a rate . [Like, “something” per “something”] </li></ul><ul><li>More particularly, the first “something” is the units of the function f(x), and the second “something” is the units of the x. </li></ul><ul><li>“ something” per “something” </li></ul>
  18. 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:
  19. 28. The word “differentiable” means… _________________________________ <ul><li>If it is differentiable at a point, then it is continuous at that point. </li></ul><ul><li>BUT </li></ul><ul><li>Just because it is continuous at a point doesn’t necessarily mean it is differentiable there. </li></ul><ul><li>Example: </li></ul>
  20. 29. What suggested HW problems to try? <ul><li>In section 2.3, try numbers 3-11 odd, 13a, 17a. We will do more of section 2.3 in the future. </li></ul><ul><li>In section 2.1, OMIT # 11, 23, 33, 43, 47, 49, 61. With #15-23, the y-value is extra information because you only use the x-coordinate. </li></ul>

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