ap calculus bc thursday, 03 september 2015 objective tsw (1) determine continuity at a point and...
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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.
Sec. 2.6: Continuity
Types of discontinuities:
Hole Gap HoleNOTE: The third type of discontinuity is an asymptote.
Sec. 2.6: Continuity
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.
Sec. 2.6: Continuity
Ex: Discuss the continuity of the following:
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.
Sec. 2.6: Continuity
Ex: Discuss the continuity of the following:
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.
Sec. 2.6: Continuity
Ex: Discuss the continuity of the following:
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
Sec. 2.6: Continuity
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
Sec. 2.6: Continuity
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
Sec. 2.6: Continuity
Functions that are continuous at every point in their domain:
a) Polynomial functions
b) Rational functions
c) Radical functions
d) Trigonometric functions
Sec. 2.6: Continuity
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
Sec. 2.6: Continuity
Ex: State why the following are continuous:
b)
f is a composition of a radical function and a polynomial
( ) 4f x x
Sec. 2.6: Continuity
Ex: State why the following are continuous:
c)
f is the product of a polynomial function and a trig function
3( ) ( 3 )(tan )f x x x x
Sec. 2.6: Continuity
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.
Sec. 2.6: Continuity
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.
Sec. 2.6: Continuity
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.
Sec. 2.6: Continuity
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