Asymptotic and Unbounded Behavior - AP Calculus BC

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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?

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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Question

The rate of growth of the population of Reindeer in Norway is proportional to the population. The population increased from 9876 to 10381 between 2013 and 2015. What is the expected population in 2030?

Answer

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:

$p(t)=p_0e^{kt}$

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:

$10381=9876e^{k(2015-2013)}$

$\frac{10381}{9876}=e^{2k}$

$2k=ln(\frac{10381}{9876})$

$k=\frac{ln(\frac{10381}{9876})}{2}=0.0252$

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

$P=10381e^{(2030-2015)(0.0252)} \approx 15150$

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Question

The rate of decrease due to poaching of the elephants in unprotected Sahara is proportional to the population. The population in one region decreased from 1038 to 817 between 2010 and 2015. What is the expected population in 2017?

Answer

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:

$p(t)=p_0e^{kt}$

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:

$817=1038e^{k(2015-2010)}$

$\frac{817}{1038}=e^{5k}$

$5k=ln(\frac{817}{1038})$

$k=\frac{ln(\frac{817}{1038})}{5}=-0.048$

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

$P=817e^{(2017-2015)(-0.048)} \approx 742$

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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?

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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Question

The rate of growth of the population of Reindeer in Norway is proportional to the population. The population increased from 9876 to 10381 between 2013 and 2015. What is the expected population in 2030?

Answer

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:

$p(t)=p_0e^{kt}$

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:

$10381=9876e^{k(2015-2013)}$

$\frac{10381}{9876}=e^{2k}$

$2k=ln(\frac{10381}{9876})$

$k=\frac{ln(\frac{10381}{9876})}{2}=0.0252$

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

$P=10381e^{(2030-2015)(0.0252)} \approx 15150$

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Question

The rate of decrease due to poaching of the elephants in unprotected Sahara is proportional to the population. The population in one region decreased from 1038 to 817 between 2010 and 2015. What is the expected population in 2017?

Answer

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:

$p(t)=p_0e^{kt}$

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:

$817=1038e^{k(2015-2010)}$

$\frac{817}{1038}=e^{5k}$

$5k=ln(\frac{817}{1038})$

$k=\frac{ln(\frac{817}{1038})}{5}=-0.048$

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

$P=817e^{(2017-2015)(-0.048)} \approx 742$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Observation of the graph shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point:}\&x=\frac{2}{17}\&\text{We find a zero denominator for the function: }\&\frac{(13x - 9)}{(17x - 2)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=-\frac{3}{8}\&\text{This matches the ratio of coefficients for the function: }\&-\frac{(6x - 18)}{(16x + 9)}\end{align}$

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Question

$\begin{align}&\text{Which of the four following functions is depicted in the graph above?}\end{align}$

Answer

$\begin{align}&\text{Observation of the graph shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point centered around:}\&x=1.3\&\text{We find a zero denominator for the function: }\&\frac{(4x + 18)}{(14x - 19)}\end{align}$

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Question

$\begin{align}&\text{Which of the four following functions is depicted in the graph above?}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=-0.8\&\text{This matches the ratio of coefficients for the function: }\&-\frac{(5x + 12)}{(6x - 10)}\end{align}$

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Question

$\begin{align}&\text{Of the following functions, which most likely represents the graph above?}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=0.3\&\text{This matches the ratio of coefficients for the function: }\&\frac{(3x - 2)}{(10x - 11)}\end{align}$

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Question

$\begin{align}&\text{Of the following functions, which most likely represents the graph above?}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=-1.12\&\text{This matches the ratio of coefficients for the function: }\&-\frac{(19x + 9)}{(17x - 5)}\end{align}$

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Question

$\begin{align}&\text{Of the following functions, which most likely represents the graph above?}\end{align}$

Answer

$\begin{align}&\text{Observation of the graph shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point centered approximately around:}\&x=-0.3\&\text{We find a zero denominator for the function: }\&-\frac{(x - 13)}{(15x + 4)}\end{align}$

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Question

$\begin{align}&\text{Of the following functions, which most likely represents the graph above?}\end{align}$

Answer

$\begin{align}&\text{Observation of the graph shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point centered approximately around:}\&x=0.1\&\text{We find a zero denominator for the function: }\&\frac{(9x - 11)}{(15x - 1)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=0.71\&\text{This matches the ratio of coefficients for the function: }\&\frac{(12x - 20)}{(17x)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=1.78\&\text{This matches the ratio of coefficients for the function: }\&\frac{(16x - 13)}{(9x - 4)}\end{align}$

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Question

$\begin{align}&\text{Of the following functions, which most likely represents the graph above?}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=0.2\&\text{This matches the ratio of coefficients for the function: }\&\frac{(3x + 14)}{(15x - 16)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=-1.67\&\text{This matches the ratio of coefficients for the function: }\&-\frac{(20x + 10)}{(12x - 10)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Observation of the graph shows that there’s a sharp increase}\&\text{and decrease on either side of a particular x-value. This type}\&\text{of behavior is observed when there’s a vertical asymptote. An}\&\text{asymptote is a value that a function may approach, but will}\&\text{never actually attain. In the case of vertical asymptotes, this}\&\text{behavior occurs if the function approaches infinity for a given}\&\text{x-value, often when a zero value appears in a denominator. Noting}\&\text{this rule, the above function has a zero in the denominator}\&\text{at a definite point:}\&x=\frac{2}{17}\&\text{We find a zero denominator for the function: }\&\frac{(13x - 9)}{(17x - 2)}\end{align}$

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Question

$\begin{align}&\text{Decide which of the following functions, most likely represents the graph above.}\end{align}$

Answer

$\begin{align}&\text{Notice that as x increases, the function graphed appears to}\&\text{level out, flattening towards a certain value. This value is}\&\text{what is known as a horizontal asymptote. An asymptote is a value}\&\text{that a function approaches, but never actually reaches. Think}\&\text{of a horizontal asymptote as the limit of a function as x approaches}\&\text{infinity. In such a case, as x approaches infinity, any constants}\&\text{added or subtracted in the numerator and denominator become}\&\text{irrelevant. What matters is the power of x in the denominator}\&\text{and the numerator; if those are the same, then the coefficients}\&\text{define the asymptote. We see that the function flattens towards:}\&f(\infty)=-\frac{3}{8}\&\text{This matches the ratio of coefficients for the function: }\&-\frac{(6x - 18)}{(16x + 9)}\end{align}$

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