AP Calculus BC Thursday, 03 September 2015 OBJECTIVE TSW (1) determine continuity at a point and continuity on an open interval; (2) determine one- sided.

AP Calculus BCThursday, 03 September 2015

• OBJECTIVE TSW (1) determine continuity at a point and continuity on an open interval; (2) determine one-sided limits and continuity on a closed interval; (3) use properties of continuity; and (4) understand and use the Intermediate Value Theorem.

• T-Shirt sales: S – XL $7.00 2XL – 4XL $9.00

– Due by Tuesday, 08 September 2015.

Sec. 2.6: Continuity


Informally, a function f is continuous at x = c if there is no “interruption” in the graph of f at x = c.

That is, the graph of f has no holes, jumps, or gaps at c.




Types of discontinuities:

Hole Gap HoleNOTE: The third type of discontinuity is an asymptote.


Ex: Discuss the continuity of the following:

) a1

f xx

f has a domain of (–∞, 0) ᴜ (0, ∞), so f is continuous at every x-value in its domain.

f has a nonremovable discontinuity (asymptote) at x = 0.



2 1

1b)

xg x

x

g has a domain of (–∞, 1) ᴜ (1, ∞), so g is continuous at every x-value in its domain.

g has a removable discontinuity (hole) at x = 1.



2

1, 0

1,)

0c

x xh x

x x

h has a domain of (–∞, ∞), so h is continuous at every x-value.

h is everywhere continuous.



d n) siy x

y has a domain of (–∞, ∞), so y is continuous at every x-value.

y is everywhere continuous

Sec. 2.6: ContinuityOne-Sided Limits

A one-sided limit considers only one side of a given value.

Ex: The limit from the right means that x approaches c from values greater than c.

A similar meaning is given to the limit from the left.

limx c

f x L

limx c

f x L


Ex: Find the following:

2

2lim 4

xx

2

20lim 4

xx


Ex: Using the definition of continuity, determine if f is continuous at x = 2.

4, 2

3 1, 2

x xf x

x x

2 6, so 2 is defin ) edi ff

2

li) imi 6x

f x

2

lim 7x

f x

2

lim DNEx

f x

is not continuous at 2.f x


Ex: Using the definition of continuity, determine if f is continuous at x = 2.

5, 2

3 1, 2

x xf x

x x

2 7, so 2 is defin ) edi ff

2

i 2 li mii) x

f f x

2

lim 7x

f x

2

lim 7x

f x

is continuous at 2.f x

2

lim existsx

f x

2

li) imi 7x

f x


The two one-sided limits must equal each other.



Functions that are continuous at every point in their domain:

a) Polynomial functions

b) Rational functions

c) Radical functions

d) Trigonometric functions




Combining these properties with continuous functions allows you to state that many functions are continuous.

Ex: State why the following are continuous:

a) f(x) = x2 + 1 – sin x

f is the sum of a polynomial and a trig function



b)

f is a composition of a radical function and a polynomial

( ) 4f x x



c)

f is the product of a polynomial function and a trig function

3( ) ( 3 )(tan )f x x x x



f is continuous on [a, b]. three c’s f(c) = k.

f is not continuous on [a, b]. no c’s f(c) = k.


Ex: Use the Intermediate Value Theorem to show that f(x) = x3 +2x – 1 has at least one zero in [0, 1].

i) f(x) is continuous (because it’s a polynomial)

ii) f(0) = –1

f(1) = 2

f(0) < 0 < f(1)

By the IVT, at least one zero in [0, 1] f(c) = 0.


Ex: (a) Verify that the Intermediate Value Theorem applies in the indicated interval for f(x). (b) Find the value of c guaranteed by the theorem.

f(x) = x2 – 6x + 8, [0, 3], f(c) = 0(a) i) f is continuous on [0, 3] (polynomial)

ii) f(0) = 8 and f(3) = –1

f(0) > f(c) = 0 > f(3)

By the IVT, at least one zero in [0, 3] f(c) = 0.


Ex: (a) Verify that the Intermediate Value Theorem applies in the indicated interval for f(x). (b) Find the value of c guaranteed by the theorem.

f(x) = x2 – 6x + 8, [0, 3], f(c) = 0(b) Find c.

x2 – 6x + 8 = 0

(x – 4)(x – 2) = 0

x = 4, x = 2

Since x = 2 is in the interval and f(2) = 0,

c = 2

AP Calculus BC Thursday, 03 September 2015 OBJECTIVE TSW (1) determine continuity at a point and continuity on an open interval; (2) determine one- sided.

Documents

continuity f

continuous functions

definition of continuity

use properties of continuity

function f

graph of f

given value

x approaches c