Connections Between Function Features And Limits

1. Which existence theorem allows use to use guess-and-check to solve equations which can be written in the form F(x) = 0 for a continuous function F?

The Intermediate Value Theorem

The Mean Value Theorem

The Extreme Value Theorem

Rolle's Theorem

The Intermediate Value Theorem states that if a continuous function has different signs at two points, then it must have at least one root (zero) between those two points. In the context of solving equations of the form F(x) = 0, this means that if we can find two points where F(x) takes on different signs, we can conclude that there is at least one solution to the equation within that interval. This allows us to use guess-and-check methods, trying different values of x within the interval, to find the solution.

Explanation

The Intermediate Value Theorem states that if a continuous function has different signs at two points, then it must have at least one root (zero) between those two points. In the context of solving equations of the form F(x) = 0, this means that if we can find two points where F(x) takes on different signs, we can conclude that there is at least one solution to the equation within that interval. This allows us to use guess-and-check methods, trying different values of x within the interval, to find the solution.

2. Match the following

The limit as x approaches plus or minus infinity is a number.

The limit as x approaches a number from the left and/or right is +/–infinity.

The limit from the left and the limit from the right both exist but don't match.

The limit exists but doesn't match the function value.

The instantaneous rate is zero.

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3. Which of the following must be true if a function f(x) does not have a minimum on an interval [a,b].

F(x) must approach negative infinity somewhere in the interval [a,b]

F(x) must have a discontinuity on [a,b]

F(a) must be greater than f(b)

F(x) must have a maximum on the interval [a,b]

If a function does not have a minimum on an interval [a, b], it means that the function either constantly increases or constantly decreases throughout the interval. In other words, there are no points where the function reaches a local minimum. Therefore, there must be a point of discontinuity in the function on the interval [a, b]. This is because if the function is continuous throughout the interval, it would have to reach a minimum at some point.

Explanation

If a function does not have a minimum on an interval [a, b], it means that the function either constantly increases or constantly decreases throughout the interval. In other words, there are no points where the function reaches a local minimum. Therefore, there must be a point of discontinuity in the function on the interval [a, b]. This is because if the function is continuous throughout the interval, it would have to reach a minimum at some point.

4. Select all the places to look for extreme points (maxima, minima):

Where the instantaneous rate of change is zero

Where the instantaneous rate of change does not exist

At the endpoints of the interval of interest

Where the function value is zero

Extreme points (maxima, minima) occur at places where the instantaneous rate of change is zero, as well as where the instantaneous rate of change does not exist. This is because at these points, the slope of the function is either flat (zero rate of change) or undefined (non-existent rate of change), indicating a potential extreme point. Additionally, extreme points can also occur at the endpoints of the interval of interest, as these are the boundaries of the function's behavior. The function value being zero does not necessarily indicate an extreme point, as it could be a point of intersection or a regular point on the graph.

Explanation

Extreme points (maxima, minima) occur at places where the instantaneous rate of change is zero, as well as where the instantaneous rate of change does not exist. This is because at these points, the slope of the function is either flat (zero rate of change) or undefined (non-existent rate of change), indicating a potential extreme point. Additionally, extreme points can also occur at the endpoints of the interval of interest, as these are the boundaries of the function's behavior. The function value being zero does not necessarily indicate an extreme point, as it could be a point of intersection or a regular point on the graph.

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5. Select all the correct statements about extreme points of the function in the graph.

The absolute minimum value is 0.

The absolute maximum value is 14/3.

There is a relative minimum between x=–5 and x=–4.

There is one point which is a relative maximum where the instantaneous rate of change is zero.

The instantaneous rate of change does not exist at x=–3 and there is a relative maximum there.

(1,1) is a relative maximum point.

The left end point is a relative maximum.

The given correct statements about extreme points of the function in the graph are:
- There is a relative minimum between x=-5 and x=-4.
- There is one point which is a relative maximum where the instantaneous rate of change is zero.
- The left end point is a relative maximum.

Explanation

The given correct statements about extreme points of the function in the graph are:
- There is a relative minimum between x=-5 and x=-4.
- There is one point which is a relative maximum where the instantaneous rate of change is zero.
- The left end point is a relative maximum.

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