The IVT is often used to verify that a function has a zero. For the following graph what would
be the proper way to state the IVT theorem to show that there is a zero in the range?
f(x) is continuous on [-5, -2] and let k be zero. Then there exists a number c such that, f(c) = 0.
f(x) is continuous on [-5, 5] and let k be zero. Then there exists a number c such that, f(c) = 0.
f(x) is continuous on [-2, 5] and let k be zero. Then there exists a number c such that, f(c) = 0.
f(x) is continuous on [4, 5] and let k be zero. Then there exists a number c such that, f(c) = 0.
Over what interval will the Intermediate Value Theorem (IVT) apply?
everywhere
Some mathematicians wanted to use the IVT as the definition of continity. i.e. if the IVT does
not hold then the function can not be continuous.
For the following function (see graph) Which statement is true?
the IVT can be applied over the range
the IVT can be applied over the range
the IVT can be applied over the range
the IVT can be applied over the whole range shown in the graph.
Some mathematicians wanted to use the IVT as the definition of continuous. i.e. if the IVT
does not hold then the function can not be continuous.For the following graph what is the
counter example that shows that the IVT does not hold over a non-continuous graph?
f (-4) = 1, f(2) = -1 so there must be a point where f(x) = .25.
f(-4) = 1, f(-2) = 1 so there must be a point where f(x) = .25.
f(4) = -1, f(2) = -1 so there must be a point where f(x) = .25.
the IVT holds for the above graph.
The IVT is often used to verify that a function has a zero. For the following graph what would
be the proper way to state the IVT theorem to show that there is a zero in the range?
f(x) is continuous on [-2, -1] and let k be zero. Then there exists a number c such that, f( c ) = 0.
f(x) is continuous on [1,2] and let k be zero. Then there exists a number c such that, f( c ) = 0.
f(x) is continuous on [-1, 2.5] and let k be any zero. Then there exists a number c such that, f( c ) = 0.
f(x) is continuous on [2.5,3] and let k be zero. Then there exists a number c such that, f( c ) = 0.
Use the IVT (Intermediate Value Theorem) to show that
has a zero.
f(x) is continuous on [0,1], f(0) < 0 , f(1) = 0, so we can use the IVT to show there is a zero in [0,1]
f(x) is continuous on [-1,2], f(-1) < 0 , f(2) > 0, so we can use the IVT to show there is a zero in [-1,2]
f(x) is continuous on [-2,1], f(-2) < 0 , f(1) = 0, so we can use the IVT to show there is a zero,
between [-2,1].
The IVT cannot be applied to find a zero.
Use the IVT (Intermediate Value Theorem) to show that f(x) = x has a zero in the interval [-3,3].
f(x) is continuous on [1,2], f(1)= 1, f(2) = 2, so we can use the IVT to show there is a zero in [1,2]
f(x) is continuous on [-1,1], f(-1)= -1, f(1) = 1, so we can use the IVT to show there is a zero in [-1,1]
f(x) is continuous on [0,2], f(0)= 0, f(2) = 2, so we can use the IVT to show there is a zero, between
[1,2].
The IVT cannot be applied to find a zero.
Use the IVT (Intermediate Value Theorem) to show that
You can use the IVT to show there is a zero in [-1,1]
You can use the IVT to show there is a zero in [-1,2]
You can use the IVT to show that there is a zero in [-1, 0]
The IVT cannot be applied to find a zero anywhere.
has a zero.
Use the IVT (Intermediate Value Theorem) to show that
has a zero.
f(x) is continuous on [0,1], f(0) < 0 , f(1) = 0, so we can use the IVT to show there is a zero in [0,1]
f(x) is continuous on [-1,2], f(-1) < 0 , f(2) > 0, so we can use the IVT to show there is a zero in [-1,2]
f(x) is continuous on [-2,0], f(-2) < 0 , f(0) = 0, so we can use the IVT to show there is a zero,
between [-2,1].
The IVT cannot be applied to find a zero.
Use the IVT (Intermediate Value Theorem) to show that
You can not use the IVT to show there is a zero in [-1,1]
You can use the IVT to show there is a zero in [-1,2]
You can use the IVT to show that there is a zero in [-1, 0]
The IVT cannot be applied to find a zero anywhere.
What is the limit of 1/x as x approaches 0 from the positive side?
What is
where
What is
where f(x) is as shown in the graph below?
0
-1
1
Does not exist
has a zero.
What is
?
-1
1
0
Limit Does Not Exist
What is
x
f(x)
0.80
5
where f(x) is as shown in the table below?
.90
.95
1
1.05
4.5
5.99
8
8.01
8
5
6
Does Not Exist
Use direct substitution to find
1
Does Not Exist
Use direct substitution to find
-5
5
0
Does Not Exist
1.10
7.15
1.2
8.25
Limit properties to evaluate
Does not exist
1
0
Use limit properties to evaluate
1
-1
0
Does Not Exist
Evaluate
Evaluate the
Evaluate
0
Does Not Exist
Evaluate
-1
1
0
Does Not Exist
Evaluate
-1
1
0
Does Not Exist
For the following graph, where does the limit of f(x) not exist?
For the following graph, which limits equal
For the following graph, what is
What are the
and
?
?
What is
0
Does Not Exist
Infinity
-infinity
What is
0
Does Not Exist
?
What does it mean to be a continuous function in simple terms?
Describe a case where f(x) has a limit but is not continuous?
Explain why
is not continuous at x = 0.
Where is f(x) discontinuous?
X=1
X=3
X=0
Everywhere
Show, using limits, that f(x) = x2 – x + 3, is continuous at x = 2.
Show, using limits, that f(x) = tan(x), is continuous at x = 0.
Use limits to determine if
is continuous at x =1
so the function is continuous at x =1
so the function is not continuous at x =1.
so the function is not continuous at x =1.
Can not determine using limits
Use limits to determine if
is continuous at x =1
so the function is continuous at x =1
so the function is not continuous at x =1.
so the function is not continuous at x =1.
Can not determine using limits
What kind of discontinuity does
have at x = 3?
Describe a function a function with a jump discontinuity.
Describe a function with an oscillating discontinuity at 0.
What kind of discontinuity does
at x = 0?
Infinite
Jump
Oscillating
No discontinuity
What kind of discontinuity does f(x) =
have at x = 0?
Infinite
Jump
Oscillating
No discontinuity
What kind of discontinuity does f(x) =
at x = 3?
Infinite
Jump
Oscillating
No discontinuity
What, if any, are the removable discontinuities of f(x) =
? How do you remove them?
How do you remove removable discontinuities?
What are the removable discontinuities of f(x) =
? How do you remove them?
Evaluate
0
1
5
Does Not Exist
Suppose
and
exist, but are not equal. Which of the following could be true
about f at x = 3?
I. f is continuous at x = 3.
II. There is a jump in the graph of f at x = 3.
III. There is a vertical asymptote in the graph of f at x = 3.
I Only
II Only
III Only
II and III Only
Give a simple example of what IVT means in real language.