Functions, Graphs, and Limits - AP Calculus BC

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Question

$\newline\vec{v} =(x^2,-x,5)\newline\vec{u} =(2,3x,11)\newline$

Calculate $\vec{v} + \vec{u}$

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Answer

Calculate the sum of vectors.

In general,

$\vec{v} = (v_1,v_2,v_3)$

$\vec{u} = (u_1,u_2,u_3)$

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2,v_3 + u_3)$

Solution:

$\newline\vec{v} =(x^2,-x,5)\newline\vec{u} =(2,3x,11)\newline$

$\vec{v}+\vec{u}=(x^2,-x,5)+(2,3x,11)$

$\vec{v}+\vec{u}=(x^2+2,-x+3x,5+11)$

$\vec{v}+\vec{u}=(x^2 + 2,2x,16)$

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Question

Calculate the polar form hypotenuse of the following cartesian equation:

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Answer

In a cartesian form, the primary parameters are and . In polar form, they are and $\theta$

is the hypotenuse, and $\theta$ is the angle created by .

2 things to know when converting from Cartesian to polar.

$x= r\cos(\theta)$

$y= r\sin(\theta)$

You want to calculate the hypotenuse,

Solution:

$y=10x^4=r\sin(\theta)$

$y=10(r\cos(\theta))^4$

$y=10r^4\cos^4(\theta) = r\sin(\theta)$

$r^3 = \frac{\tan(\theta)}{10\cos^3(\theta)}$

$r = \sqrt[3]{\frac{\tan(\theta)}{10\cos^3(\theta)}}$

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Question

Find the vector form of to .

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Answer

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given and

$\overrightarrow{v}=[d-a, e-b, f-c]$

In our case we have ending point at and our starting point at .

Therefore we would set up the following and simplify.

$\overrightarrow{v}=[6-0,3-1,1-3]=[6,2,-2]$

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Question

Given the above graph of , what is $\lim_{x\rightarrow 0}f(x)$ ?

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Answer

Examining the graph, we can observe that $\lim_{x\rightarrow 0}f(x)$ 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 :

A value exists in the domain of

The limit of exists as approaches

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 $\lim_{x\rightarrow 0}f(x)$ approaches two different limits: $\infty$ from the left and $-\infty$ from the right.

Based on the above, condition #3 is also not satisfied because $\lim_{x\rightarrow 0}f(x)$ is not equal to the multiple values of .

Thus, $\lim_{x\rightarrow 0}f(x)$ does not exist.

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Question

Given the graph of above, what is $\lim_{x\rightarrow 0}f(x)$ ?

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Answer

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

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

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

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 $\lim_{x\rightarrow 0}f(x)=0$ as approaches .

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Question

Given the graph of above, what is $\lim_{x\rightarrow 0}f(x)$ ?

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Answer

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

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

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

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

Therefore, we can determine that $\lim_{x\rightarrow 0}f(x)$ does not exist, since approaches two different limits from either side : $-\infty$ from the left and $\infty$ from the right.

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Question

Given the above graph of , what is $\lim_{x\rightarrow 0^{+}}f(x)$ ?

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Answer

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 $\lim_{x\rightarrow 0^{+}}f(x)=\infty$ as approaches from the right.

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align}$

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }0\end{align}$

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Question

$\begin{align}&\text{Which of the values most likely represents the value of f(x) at}\&x=\infty\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(\infty)\text{ is most likely }-5\end{align}$

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Question

$\begin{align}&\text{Which of the values most likely represents the value of f(x) at}\&x=\infty\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(\infty)\text{ is most likely }-\infty\end{align}$

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Question

$\begin{align}&\text{Based on the graph, decide which of the following functions values likely depicts}\&f(-\infty)\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align}$

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }\infty\end{align}$

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Question

$\begin{align}&\text{Which of the values most likely represents the value of f(x) at}\&x=\infty\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(\infty)\text{ is most likely }\text{Nonexistent.}\end{align}$

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }\text{Nonexistent.}\end{align}$

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }0\end{align}$

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Question

$\begin{align}&\text{What is the likely value of}\&f(-\infty)\&\text{Based on the graph?}\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(-\infty)\text{ is most likely }\frac{1}{4}\end{align}$

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Question

$\begin{align}&\text{Which of the values most likely represents the value of f(x) at}\&x=\infty\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(\infty)\text{ is most likely }-\frac{1331}{512}\end{align}$

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Question

$\begin{align}&\text{Based on the graph, decide which of the following functions values likely depicts}\&f(\infty)\end{align}$

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Answer

$\begin{align}&\text{Although actually observing a function at an infinite value}\&\text{of x is impossible, we can often determine via observation if}\&\text{the function approaches a certain limit. Looking at the values}\&\text{the function approaches as the absolute value of x increases}\&\text{gives a clue. Things to look out for are if the function continuously}\&\text{climbs/decreases, flattens, or has unchecked oscillatory behavior.}\&\text{If the function continuously climbs/decreases, it likely has}\&\text{a limit that approaches positive/negative infinity. If it flattens,}\&\text{the limit is a real number. If it oscillates without control,}\&\text{the limit does not exist.}\&\text{The limit of }f(\infty)\text{ is most likely }0\end{align}$

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Question

A cylinder of height and radius is expanding. The radius increases at a rate of $0.1\frac{in}{s}$ and its height increases at a rate of $0.2\frac{in}{s}$ . What is the rate of growth of its surface area?

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Answer

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

$A=2\pi r^2+2\pi rh$

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

$\frac{dA}{dt}=4\pi r \frac{dr}{dt}+2\pi h \frac{dr}{dt} + 2\pi r \frac{dh}{dt}$

Therefore, the rate of growth of surface area is:

$\frac{dA}{dt}=4\pi (1in)\left(0.1\frac{in}{s}\right)+2\pi(1in)(4in)\left(0.1\frac{in}{s}\right)+2\pi(1in)\left(0.2\frac{in}{s}\right)$

$\frac{dA}{dt}=(0.4\pi+0.8\pi+0.4\pi)\frac{in^2}{s}=1.6\pi\frac{in^2}{s}$

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