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Section 1.1  A Preview of Calculus What is it good for, anyway? Look at p. 43 - 44 for some examples
Calculus makes the concepts of precalculus  DYNAMIC! PRECALCULUS LIMITING PROCESS CALCULUS
There are two big questions in Calculus! What is the equation of the tangent line that touches a curve at a particular point?  What is the area underneath a curve?
To get the equation of a tangent line, we must know a point and a slope.  How do we find the slope at a given point?
We can find slope if we know two points!
We can define slope for lines, how can that carry over to curves? ,[object Object],[object Object]
P(c, f(c)) is the point of tangency.  Q(c+ Δ x,  f(c+  Δ x)) is nearby point
What is the slope of the secant line? or
Here is a great site to see what is happening! ,[object Object]
[object Object],[object Object],[object Object],[object Object]
[object Object]
How do you find area under a curve? Fig. 1.3, p.46
Let’s divide it into rectangles that approximate the area!
So, how wide are the rectangles?  How tall are they? If we divide into 4 rectangles, the width would be  The height would be the  y-value at the left or right side of rectangle, or interval.
Rect. 1 Rect 2 Rect 3 Rect 4 Width .25 .25 .25 .25 Height f(  )= f(  )= f(  )= f(  )= Area = W ٠ H
If rectangles formed from right of interval: .25 ٠ 0 + .25 ٠ .0625 + .25 ٠ .25 + .25 ٠ .5625 = 0  +  .01563  +  .0625  +  .14063  =  .21876  If rectangles formed from left of interval: .25 ٠ .0625 + .25 ٠ .25 + .25 ٠ .5625 + .25  ٠ 1 = .01563  +  .0625  +  .14063  + .25  = .46876 The first is an underestimate, the second is an overestimate.  The actual value is 0.3333…
What would happen if we increased the number of rectangles? http://www.math.psu.edu/dlittle/java/calculus/area.html   p. 47/ 1-9 odd

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Calc 1.1

  • 1. Section 1.1 A Preview of Calculus What is it good for, anyway? Look at p. 43 - 44 for some examples
  • 2. Calculus makes the concepts of precalculus DYNAMIC! PRECALCULUS LIMITING PROCESS CALCULUS
  • 3. There are two big questions in Calculus! What is the equation of the tangent line that touches a curve at a particular point? What is the area underneath a curve?
  • 4. To get the equation of a tangent line, we must know a point and a slope. How do we find the slope at a given point?
  • 5. We can find slope if we know two points!
  • 6.
  • 7. P(c, f(c)) is the point of tangency. Q(c+ Δ x, f(c+ Δ x)) is nearby point
  • 8. What is the slope of the secant line? or
  • 9.
  • 10.
  • 11.
  • 12. How do you find area under a curve? Fig. 1.3, p.46
  • 13. Let’s divide it into rectangles that approximate the area!
  • 14. So, how wide are the rectangles? How tall are they? If we divide into 4 rectangles, the width would be The height would be the y-value at the left or right side of rectangle, or interval.
  • 15. Rect. 1 Rect 2 Rect 3 Rect 4 Width .25 .25 .25 .25 Height f( )= f( )= f( )= f( )= Area = W ٠ H
  • 16. If rectangles formed from right of interval: .25 ٠ 0 + .25 ٠ .0625 + .25 ٠ .25 + .25 ٠ .5625 = 0 + .01563 + .0625 + .14063 = .21876 If rectangles formed from left of interval: .25 ٠ .0625 + .25 ٠ .25 + .25 ٠ .5625 + .25 ٠ 1 = .01563 + .0625 + .14063 + .25 = .46876 The first is an underestimate, the second is an overestimate. The actual value is 0.3333…
  • 17. What would happen if we increased the number of rectangles? http://www.math.psu.edu/dlittle/java/calculus/area.html p. 47/ 1-9 odd

Hinweis der Redaktion

  1. Put in x^2 for f(x), 1for a, 2 for b
  2. Slope is (4-1)/(2-1) = 3 (1,1) y – 1 = 3(x – 1) which becomes y = 3x – 3 + 1 which is y = 3x – 2
  3. The secant line should closer to the slope line Look at website and slide point closer Show tangent line and secant line at same time
  4. Switch to e-instruction and shade in rectangles