Asymptotic and Unbounded Behavior

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AP Calculus BC › Asymptotic and Unbounded Behavior

Questions 1 - 10

Asymptoteplot

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

$-\frac{(20x + 10)}{(12x - 10)}$

$-\frac{(12x - 10)}{(20x + 10)}$

$\frac{(20x + 10)}{(12x - 10)}$

$\frac{(12x - 10)}{(20x + 10)}$

Explanation

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

Asymptoteplot

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

$-\frac{(20x + 10)}{(12x - 10)}$

$-\frac{(12x - 10)}{(20x + 10)}$

$\frac{(20x + 10)}{(12x - 10)}$

$\frac{(12x - 10)}{(20x + 10)}$

Explanation

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?

Explanation

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:

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

Asymptoteplot

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

$\frac{(12x - 20)}{(17x)}$

$\frac{(17x)}{(12x - 20)}$

$-\frac{(12x - 20)}{(17x)}$

$-\frac{(17x)}{(12x - 20)}$

Explanation

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

Asymptoteplot

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

$\frac{(12x - 20)}{(17x)}$

$\frac{(17x)}{(12x - 20)}$

$-\frac{(12x - 20)}{(17x)}$

$-\frac{(17x)}{(12x - 20)}$

Explanation

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

Explanation

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:

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

Asymptoteplot

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

$-\frac{(19x + 9)}{(17x - 5)}$

$-\frac{(17x - 5)}{(19x + 9)}$

$\frac{(19x + 9)}{(17x - 5)}$

$\frac{(17x - 5)}{(19x + 9)}$

Explanation

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

Asymptoteplot

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

$-\frac{(19x + 9)}{(17x - 5)}$

$-\frac{(17x - 5)}{(19x + 9)}$

$\frac{(19x + 9)}{(17x - 5)}$

$\frac{(17x - 5)}{(19x + 9)}$

Explanation

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

Asymptoteplot

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

$\frac{(16x - 13)}{(9x - 4)}$

$\frac{(9x - 4)}{(16x - 13)}$

$-\frac{(16x - 13)}{(9x - 4)}$

$-\frac{(9x - 4)}{(16x - 13)}$

Explanation

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

Asymptoteplot

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

$\frac{(16x - 13)}{(9x - 4)}$

$\frac{(9x - 4)}{(16x - 13)}$

$-\frac{(16x - 13)}{(9x - 4)}$

$-\frac{(9x - 4)}{(16x - 13)}$

Explanation

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

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