Your SlideShare is downloading. ×
0
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Antiderivatives and indefinite integration2009
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Antiderivatives and indefinite integration2009

233

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
233
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
6
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Drive activity • Please submit your drive activities.
  • 2. Sec. 4.1: Antiderivatives
  • 3. Uses of integration 1. 2. 3. 4. 5. “Undoes” differentiation. Finds the area under a curve. Finds the volume of a solid. Finds the center of mass. Finds s(t) given a(t) or v(t). *** and many more….
  • 4. The Uniqueness of Antiderivatives 2 Suppose f x 3x , find an antiderivative of f. That is, 2 find a function F(x) such that F ' x 3x . F x x 3 Using the Power Rule in Reverse Is this the only function whose derivative is 3x2? H x H' x x 3 5 3x 2 K x x 3 11 3x 2 K' x M x M' x x 3 3x 2 There are infinite functions whose derivative is 3x2 whose general form is: C is a constant real G x 3 x C number (parameter)
  • 5. When we find antiderivatives and add the constant C, we are creating a family of curves for each value of C. x3 C C=0 C=1 C=3
  • 6. Indefinite Integral f ( x)dx Integral Sign Integrand F ( x) C The Indefinite Integral Variable of Integration The constant of Integration
  • 7. Indefinite Integral The indefinite integral gives a family of functions! (Not a value) The indefinite integral always has a constant! Vs. The definite integral (later) gives a numerical value.
  • 8. Summary Integration is the “inverse” of differentiation. d F ( x) dx dx F ( x) C Differentiation is the “inverse “ of integration. d dx f ( x)dx f ( x)
  • 9. Basic Integration Rules 0dx C kdx kx C k f ( x )dx k f ( x )dx
  • 10. Basic Integration Rules f ( x ) g ( x ) dx f ( x )dx n 1 n x dx x C n 1 Power Rule g ( x )dx
  • 11. Basic Integration Rules cos xdx sin xdx 2 sec xdx sin x C cos x C tan x C
  • 12. Basic Integration Rules 2 csc xdx sec x tan xdx csc x cot xdx cot x C sec x C csc x C
  • 13. Examples Find each of the following indefinite integrals. 5 a. x dx b. 5 x3 dx x sin x dx c. 1 6 d. 1 x 6 C cos x C 5 4 x4 C dx 2 x C
  • 14. Application problem • A ball is thrown upward with an initial velocity of 64 ft/ sec from an initial height of 80 feet. a. Find the position function, s(t) b. When does the ball hit the ground?
  • 15. Example A particle moves along a coordinate axis in such a way that 3 its acceleration is modeled by a t 2t for time t > 0. If the particle is at s = 5 when t = 1 and has velocity v = - 2 at this time, where is it when t = 4? Integrate the acceleration to find velocity: v t 2t 3 dt 2 t 3 1 3 1 2 dt t 3 1 t 2 t C 2 C Use the Initial Condition to find C for velocity: 2 1 2 C C v t 1 1 Integrate the Velocity to find position: s t t 5 1 2 1 Answer the Question: 1 dt t 2 dt 1 2 1 1 dt t 2 1 1t C t Use the Initial Condition to find C for position: 1 C s 4 C 4 1 4 5 5 1.25 s t t 1 1 t C t 5

×