1 / 20

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus. Finally!. Objective…. To integrate using the Fundamental Thm of Calc. Pandora’s box…. Fundamental Thms. The Fundamental Theorem of Arithmetic: Any positive integer can be represented in exactly one way as a product of primes.

Télécharger la présentation

Fundamental Theorem of Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Fundamental Theorem of Calculus Finally!

  2. Objective… • To integrate using the Fundamental Thm of Calc

  3. Pandora’s box…

  4. Fundamental Thms • The Fundamental Theorem of Arithmetic: • Any positive integer can be represented in exactly one way as a product of primes. • The Fundamental Theorem of Algebra: • Every polynomial of degree n has exactly n zeroes. • The Fundamental Theorem of Geometry: • No theorem wears this title, but perhaps the Pythagorean Theorem deserves it.

  5. Integrals… area under the curve • No problem if it’s a geometric shape… (4.3) • What if it’s not? How could we find the area under the curve?

  6. Rectangles…

  7. An easier example…. • This is called Riemann Sums • Using left-hand endpoints with 4 rectangles • Area =

  8. What if…. • We use right-hand endpoints and 4 rectangles? • Area =

  9. What’s a more accurate way to find area?

  10. How many rectangles is the best? f(x) = y- value or height and Δx = (b-a)/n (n is the number of rectangles)

  11. Riemann Sums and definite integrals

  12. Fundamental Theorem of Calculus • If f is cont on [a,b] and F is an antiderivative of f on [a,b] then

  13. Example

  14. What about a + C?

  15. Absolute values…

  16. A different example • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2. • Step 1… draw graph

  17. Ex cont… • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2. • Step 2: Write the integral and integrate

  18. Average Value of a function • Average value = Find the average value of f(x) = 3x^2 – 2x on [1,4]

  19. Pg 283, #31 • A company purchases a new machine for which the rate of depreciation is dV/dt = 10,000(t-6) where 0< t< 5 and V is the value of the machine after t years. What is the total loss of value of the machine over the first 3 years?

More Related