Post on 28-Dec-2015
Five-Minute Check
Then/Now
New Vocabulary
Key Concept: Limits
Key Concept: Types of Discontinuity
Concept Summary: Continuity Test
Example 1: Identify a Point of Continuity
Example 2: Identify a Point of Discontinuity
Key Concept: Intermediate Value Theorem
Example 3: Approximate Zeros
Example 4: Graphs that Approach Infinity
Example 5: Graphs that Approach a Specific Value
Example 6: Real-World Example: Apply End Behavior
Use the graph of f (x) to find the domain and range of the function.
A. D = , R =
B. D = , R = [–5, 5]
C. D = (–3, 4) , R = (–5, 5)
D. D = [–3, 4], R = [–5, 5]
Use the graph of f (x) to find the y-intercept and zeros. Then find these values algebraically.
A. y-intercept = 9, zeros: 2 and 3
B. y-intercept = 8, zeros: 1.5 and 3
C. y-intercept = 9, zeros: 1.5 and 3
D. y-intercept = 8, zero: –1
Use the graph of y = –x 2 to test for symmetry with
respect to the x-axis, y-axis, and the origin.
A. y-axis
B. x-axis
C. origin
D. x- and y-axis
You found domain and range using the graph of a function. (Lesson 1-2)
• Use limits to determine the continuity of a function, and apply the Intermediate Value Theorem to continuous functions.
• Use limits to describe end behavior of functions.
• continuous function
• limit
• discontinuous function
• infinite discontinuity
• jump discontinuity
• removable discontinuity
• nonremovable discontinuity
• end behavior
Identify a Point of Continuity
Check the three conditions in the continuity test.
Determine whether is continuous at
. Justify using the continuity test.
Because , the function is defined at
1. Does exist?
Identify a Point of Continuity
2. Does exist?
Construct a table that shows values of f(x) approaching from the left and from the right.
The pattern of outputs suggests that as the value
of x gets close to from the left and from the right,
f(x) gets closer to . So we estimate that
.
Identify a Point of Continuity
3. Does ?
Because is estimated to be and
we conclude that f (x) is continuous at . The
graph of f (x) below supports this conclusion.
Identify a Point of Continuity
Answer: 1.
2. exists.
3. .
f (x) is continuous at .
Determine whether the function f (x) = x 2 + 2x – 3 is
continuous at x = 1. Justify using the continuity test.
A. continuous; f (1)
B. Discontinuous; the function is undefined at x = 1
because does not exist.
Identify a Point of Discontinuity
A. Determine whether the function is
continuous at x = 1. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
1. Because, is undefined, f (1) does not exist.
Identify a Point of Discontinuity
2. Investigate function values close to f(1).
The pattern of outputs suggests that for values of x approaching 1 from the left, f (x) becomes increasingly more negative. For values of x approaching 1 from the right, f (x) becomes increasing more positive.
Therefore, does not exist.
Identify a Point of Discontinuity
Answer: f (x) has an infinite discontinuity at x = 1.
3. Because f (x) decreases without bound as x approaches 1 from the left and f (x) increases without bound as x approaches 1 from the right, f (x) has infinite discontinuity at x = 1. The graph of f (x) supports this conclusion.
Identify a Point of Discontinuity
B. Determine whether the function is
continuous at x = 2. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
1. Because, is undefined, f (2) does not exist.
Therefore f (x) is discontinuous at x = 2.
Identify a Point of Discontinuity
2. Investigate function values close to f (2).
The pattern of outputs suggests that f (x)
approaches 0.25 as x approaches 2 from each
side, so .
Identify a Point of Discontinuity
3. Because exists, but f (2) is undefined,
f (x) has a removable discontinuity at x = 2. The
graph of f (x) supports this conclusion.
Answer: f (x) is not continuous at x = 2, with a removable discontinuity.
A. f (x) is continuous at x = 1.
B. infinite discontinuity
C. jump discontinuity
D. removable discontinuity
Determine whether the function is
continuous at x = 1. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
Approximate Zeros
Investigate function values on the interval [2, 2].
A. Determine between which consecutive integers
the real zeros of are located on the
interval [–2, 2].
Approximate Zeros
Answer: There are two zeros on the interval, –1 < x < 0 and 1 < x < 2.
Because f (1) is positive and f
(0) is negative, by the Location Principle, f (x) has a zero between 1 and 0. The value of f (x) also changes sign for [1,2]. This indicates the existence of real zeros in each of these intervals. The graph of f (x) supports this conclusion.
Approximate Zeros
B. Determine between which consecutive integers the real zeros of f (x) = x
3 + 2x + 5 are located on the interval [–2, 2].
Investigate function values on the interval [–2, 2].
Answer: –2 < x < –1.
Approximate Zeros
Because f (2) is negative and f (–1) is positive, by the Location Principle, f (x) has a zero between –2 and –1. This indicates the existence of real zeros on this interval. The graph of f (x) supports this conclusion.
A. Determine between which consecutive integers the real zeros of f (x) = x
3 + 2x 2 – x – 1 are located
on the interval [–4, 4].
A. –1 < x < 0
B. –3 < x < –2 and –1 < x < 0
C. –3 < x < –2 and 0 < x < 1
D. –3 < x < –2, –1 < x < 0, and 0 < x < 1
B. Determine between which consecutive integers the real zeros of f (x) = 3x
3 – 2x 2 + 3 are located on
the interval [–2, 2].
A. –2 < x < –1
B. –1 < x < 0
C. 0 < x < 1
D. 1 < x < 2
Graphs that Approach Infinity
Use the graph of f(x) = x 3 – x
2 – 4x + 4 to describe its end behavior. Support the conjecture numerically.
Graphs that Approach Infinity
Analyze Graphically
Support Numerically
Construct a table of values to investigate function values as |x| increases. That is, investigate the value of f (x) as the value of x becomes greater and greater or more and more negative.
In the graph of f (x), it appears that and
Graphs that Approach Infinity
The pattern of output suggests that as x approaches –∞, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞.
Answer:
Use the graph of f (x) = x
3 + x 2 – 2x + 1 to
describe its end behavior. Support the conjecture numerically.
A.
B.
C.
D.
Graphs that Approach a Specific Value
Use the graph of to describe its end
behavior. Support the conjecture numerically.
Graphs that Approach a Specific Value
Analyze Graphically
Support Numerically
In the graph of f (x), it appears that
.
As . As . This supports our conjecture.
Use the graph of to describe its end
behavior. Support the conjecture numerically.
A.
B.
C.
D.
Apply End Behavior
PHYSICS The symmetric energy function is
. If the y-value is held constant, what
happens to the value of symmetric energy when
the x-value approaches negative infinity?
We are asked to describe the end behavior of E (x) for
small values of x when y is held constant. That is, we
are asked to find .
Apply End Behavior
Because y is a constant value, for decreasing values
of x, the fraction will become larger and
larger, so . Therefore, as the x-value gets
smaller and smaller, the symmetric energy
approaches the value
Answer:
PHYSICS The illumination E of a light bulb is
given by , where I is the intensity and d is
the distance in meters to the light bulb. If the
intensity of a 100-watt bulb, measured in candelas
(cd), is 130 cd, what happens to the value of E
when the d-value approaches infinity?
A.
B.
C.
D.