Summary
Summary Fundamental Theorem of Calculus
- Course
- Institution
Explains the fundamental theorem of calculus and provides well-explained examples and physical interpretations of theorems
[Show more]
Seller
Follow
dazyskiies
Content preview
1.3 Fundamental Theorem of CalculusTheorem. (1.3.1):letto be a continuous function on [a,b]
i) Let
g(x) St fdt,
=
where a is a constant
then f(x)
g'(x) is
ii) Let I
be any antiderivative of of
e. FP-dx
↓
F(b)
=
-
F()
b
Notation:F(x) F(b)
=
-
F(a)
a "evaluated from a to-b
(ii)
eg. of
(Xdx
①
iff(x) x,F(X) x-
=
=
F(b) F(a)
b2 1az
- =
-
② SoexdX
F(X) 2x =
F(0) e 10
F(X)/: F(1) e 1
= - =
-
-
=
physical interpretation of(ii)
(bg'(X)dX g(x)p g(b) =
= -
g(a)
m a
m
integral rate of total change ing
change of 'g' over over [a,b]
[a,b]
a proofofFTC:(just for f(x)20)
i) set
g(x) Sof(t)dt=
img(x+h)- ga
11 we want:
g.(x) = e
,,,iy
=
yy
*
a i
-
A g(4)
=
vonisNcglyta
#
e M.h
m.n g(x
=
+
1) -
g(x) =
where m and M are the min and max values
of on
f
[x,x+h]
(for his)
by dividing by h m
=g(x h) g(x) M
+ -
=
as h ->0 m,M f(x) =
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller dazyskiies. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.09. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75323 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now