{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# BCH1(2) - Proof Observe that f is continuous on[0 π&...

This preview shows pages 1–15. Sign up to view the full content.

Continuity Hole Jump

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Pole
f is continuous at c

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Continuity
Is f continuous at ‘ p ’?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A function f : D is continuous on D if f is continuous at each point in D . The following functions are continuous on : polynomials , sin x , cos x , | x | , . The rational function p ( x ) / q ( x ) is continuous on D = { x in q ( x ) 0}. The function ln x is continuous on (0, ). x e
The Intermediate Value Thm ( IVT ) Let f be a continuous function on [ a , b ] s.t. f ( a ) f ( b ). Then for each z with f ( a ) z f ( b ), there exists c in [ a , b ] s.t. f ( c ) = z . Note. If f ( a ) 0 f ( b ), then f ( x ) = 0 has a solution in [ a , b ] .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Problem . Let f ( x ) = x ² + 1 + sin x . Prove that there exists c in [0, π ] s.t. f ( c ) = 10 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Problem . Let f ( x ) = x ² + 1 + sin x . Prove that there exists c in [0, π ] s.t. f ( c ) = 10

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . Proof. Observe that f is continuous on [0, π ], & f (0) = 1 < 10 < π ² + 1 = f ( π ). By IVT , there exists c in [0, π ] s.t. f ( c ) = 10 .  Problem Let f : [0, 1] → be a continuous function & 0 ≤ f ( x ) ≤ 1 . Show that there exists c in [0, 1] s.t. f ( c ) = c . Summary • Functions ― domain & range • Operations on functions ― compositions • Limits (left- & right-) Limits involving infinity 0 if p < q = a / b if p = q ∞ if p > q q q q p p p x b x b x b a x a x a + ⋅ ⋅ ⋅ + + + ⋅ ⋅ ⋅ + + − − ∞ → 1 1 1 1 lim • The Sandwich Theorem • Continuity...
View Full Document

{[ snackBarMessage ]}

### Page1 / 15

BCH1(2) - Proof Observe that f is continuous on[0 π&...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online