Calc 1.1

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Preview of Calculus

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  • Put in x^2 for f(x), 1for a, 2 for b
  • 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
  • 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
  • Switch to e-instruction and shade in rectangles
  • Calc 1.1

    1. 1. Section 1.1 A Preview of Calculus What is it good for, anyway? Look at p. 43 - 44 for some examples
    2. 2. Calculus makes the concepts of precalculus DYNAMIC! PRECALCULUS LIMITING PROCESS CALCULUS
    3. 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. 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. 5. We can find slope if we know two points!
    6. 6. We can define slope for lines, how can that carry over to curves? <ul><li>We can approximate the slope of a curve at a given point by finding the slope of a secant line using that point and one close by. </li></ul><ul><li>Look at figure 1.2 on p. 45 in book. </li></ul>
    7. 7. P(c, f(c)) is the point of tangency. Q(c+ Δ x, f(c+ Δ x)) is nearby point
    8. 8. What is the slope of the secant line? or
    9. 9. Here is a great site to see what is happening! <ul><li>http://www.math.psu.edu/dlittle/java/calculus/secantlines.html </li></ul>
    10. 10. <ul><li>What did you come up with for slope? </li></ul><ul><li>What is the point of tangency? </li></ul><ul><li>What is the equation of the tangent line? Hint: Remember point slope formula for an equation of a line: Given slope m and point (x 1 , y 1 ): </li></ul><ul><li>(y – y 1 ) = m(x – x 1 ) </li></ul>
    11. 11. <ul><li>If you got even closer to the point by letting Δ x get really small, what would happen? </li></ul>
    12. 12. How do you find area under a curve? Fig. 1.3, p.46
    13. 13. Let’s divide it into rectangles that approximate the area!
    14. 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. 15. Rect. 1 Rect 2 Rect 3 Rect 4 Width .25 .25 .25 .25 Height f( )= f( )= f( )= f( )= Area = W ٠ H
    16. 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. 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

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