Functions, Graphs, and Limits - AP Calculus BC

In general:

If ,

then

Derivative rules that will be needed here:

• Taking a derivative on a term, or using the power rule, can be done by doing the following:
• Special rule when differentiating an exponential function: , where k is a constant.

In this problem,

Put it all together to get

Calculate the sum of vectors.

In general,

Solution:

In order to derive the vector form of the distance between two points, we must find the difference between the , , and elements of the points.

That is, for any point

and ,

the distance is the vector

.

Subbing in our original points and , we get:

In order to derive the vector form of the distance between two points, we must find the difference between the , , and elements of the points.

That is, for any point and , the distance is the vector .

Subbing in our original points and , we get:

We can find the graph of in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

We can find the graph of in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

Examining the graph, we can observe that ![](https://lh6.googleusercontent.com/P7heoexCy1S2HIhWAOx8Kt90yltzmpA2e24vxskSGXbYz4X66XU517PqIh8y_1cZgUXNoyW43HQXEXAqDLWXQZ28yplgG4xEn-era5ygRKWN69mZ3J1xswTsDa-KLrq4LJM8QX8 $"\lim_{x\rightarrow 0}$f(x)") 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 ![](https://lh5.googleusercontent.com/UwmCajmpQYMZ9A-uvKxXR7mwucumTdrfGyS_XMzSzXtv-2vSQF15bUpKZ3hY8x5QcHEIAaohKG599Vekzb2yUL4cHaFZWIcvMEdMIdttxv5Knocr_XPCmxSXxAkPWUf6fMdJWto $"\lim_{x\rightarrow 0}$f(x)") approaches two different limits: from the left and from the right.

Based on the above, condition #3 is also not satisfied because ![](https://lh6.googleusercontent.com/zvax84PofCymRxlIiFSo2th1IluK7evqYR30rOjbPEZw6IoSPWn4H9kCqI5eZPFGdgElnT5pNdzQaa8FAV77oQa8o34n65IFAo-slniLLPPSfpU2Ibfl_Zc3fs_GsV5mOjHgFBg $"\lim_{x\rightarrow 0}$f(x)") is not equal to the multiple values of .

Thus, ![](https://lh3.googleusercontent.com/GQF7dNkHBIBB7bdbVXMiBgzwsYSd6SAkrC5QIINNrDYIMSgDGHCMFNbRV0k2CLQot3KB_vBJLC28c5xpWbfslV1fgK7np2lXISPcI2YC_b441xpLYq02RcIrpwPcsI_3dhpyeGg $"\lim_{x\rightarrow 0}$f(x)") does not exist.

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.