Functions, Graphs, and Limits - AP Calculus BC

The limit we are given is one sided, meaning we are approaching our x value from one side; in this case, the negative sign exponent indicates that we are approaching 3 from the left side, or using values slightly less than three on approach.

This corresponds to the part of the piecewise function for values less than 3. When we substitute our x value being approached, we get

The surface area of a cylinder is given by the formula:

To find the rate of growth over time, take the derivative of each side with respect to time:

Therefore, the rate of growth of surface area is:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

Where is an initial population value, and is the constant of proportionality.

Since the population increased from 9876 to 10381 between 2013 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2030:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

Where is an initial population value, and is the constant of proportionality.

Since the population decreased from 1038 to 817 between 2010 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2017:

The surface area of a cylinder is given by the formula:

To find the rate of growth over time, take the derivative of each side with respect to time:

Therefore, the rate of growth of surface area is:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

Where is an initial population value, and is the constant of proportionality.

Since the population increased from 9876 to 10381 between 2013 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2030:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

Where is an initial population value, and is the constant of proportionality.

Since the population decreased from 1038 to 817 between 2010 and 2015, we can solve for this constant of proportionality:

Using this, we can calculate the expected value from 2015 to 2017:

For a function to be continuous at a point , must exist and .

This is true for all values of except and .

Therefore, the interval of continuity is .

The piecewise function

indicates that is one when is less than five, and is zero if the variable is greater than five. At , there is a hole at the end of the split.

The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as approaches negative five. From the right, the limit approaches zero as approaches negative five.

Since the limits do not coincide, the limit does not exist for .

For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.

First, notice that

.

This means that the function is continuous everywhere.

Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.

Thus, the limit as x approaches three exists and is equal to , so I and II are true statements.

For a function to be continuous at a point , must exist and .

This is true for all values of except and .

Therefore, the interval of continuity is .

The piecewise function

indicates that is one when is less than five, and is zero if the variable is greater than five. At , there is a hole at the end of the split.

The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as approaches negative five. From the right, the limit approaches zero as approaches negative five.

Since the limits do not coincide, the limit does not exist for .

For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.

First, notice that

.

This means that the function is continuous everywhere.

Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.

Thus, the limit as x approaches three exists and is equal to , so I and II are true statements.

The limit indicates that we are trying to find the value of the limit as approaches to zero from the right side of the graph.

From right to left approaching , the limit approaches to 1 even though the value at of the piecewise function does not exist.

The answer is .

Examining the graph of the function above, we need to look at three things:

1. What is the limit of the function as it approaches zero from the left?

2. What is the limit of the function as it approaches zero from the right?

3. What is the function value at zero and is it equal to the first two statements?

If we look at the graph we see that as approaches zero from the left the values approach zero as well. This is also true if we look the values as approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.

Therefore, we can observe that as approaches .

Examining the graph above, we need to look at three things:

1. What is the limit of the function as approaches zero from the left?

2. What is the limit of the function as approaches zero from the right?

3. What is the function value as and is it the same as the result from statement one and two?

Therefore, we can determine that does not exist, since approaches two different limits from either side : from the left and from the right.

Examining the graph, we want to find where the graph tends to as it approaches zero from the right hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the right the function values of the graph tend towards positive infinity.

Therefore, we can observe that as approaches from the right.

Examining the graph, we can observe that does not exist, as is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

1. A value exists in the domain of
2. The limit of exists as approaches
3. The limit of at is equal to

Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .

We can also see that condition #2 is not satisfied because approaches two different limits: from the left and from the right.

Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .

Thus, does not exist.