Exponential Functions with Base e

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Questions and Answers

What is the approximate value of the mathematical constant 'e'?

2.718 (correct)

9.81

3.141

1.618

If $P(t) = 9 \cdot (1.02)^t$ models insect population growth, what does the '9' represent?

The time in months

The population after t months

The growth rate per month

The initial population (in thousands) (correct)

Which function represents continuous exponential growth?

$P(t) = a \cdot b^t$

$Q(t) = a \cdot t^b$

$R(t) = a + bt$

$Q(t) = a \cdot e^{kt}$ (correct)

What is the role of 't' in the functions $P(t) = 9 \cdot (1.02)^t$ and $Q(t) = 9 \cdot e^{(0.02t)}$?

Represents the time in months since the study began (A) Signup and view all the answers

Which expression is equivalent to $e^{2x}$?

$(e^x)^2$ (D) Signup and view all the answers

In the exponential growth model $Q(t) = a \cdot e^{(0.02t)}$, what does the value 0.02 represent?

The continuous growth rate (B) Signup and view all the answers

How does the value of 'a' affect the graph of the exponential function $P(t) = a \cdot e^{kt}$?

It represents the vertical intercept (initial value). (B) Signup and view all the answers

Which statement accurately compares the models $P(t) = a \cdot (1 + r)^t$ and $Q(t) = a \cdot e^{kt}$?

$Q(t)$ models continuous growth, whereas $P(t)$ models discrete growth. (D) Signup and view all the answers

Which of the following is a valid step in solving $5 \cdot e^{3a} = 90$?

Divide both sides by 5 to get $e^{3a} = 18$, then take the natural logarithm of both sides. (B) Signup and view all the answers

If $e^m = 20$, which of the following is the correct solution for $m$?

$m = \ln 20$ (A) Signup and view all the answers

Solve for $x$: $10^x = 10,000$

$x = \log_{10}(10,000)$ (B) Signup and view all the answers

What is the value of $x$ in the equation $5 \cdot 10^x = 500$?

$x = 2$ (D) Signup and view all the answers

What is the value of $x$ in the equation $10(x+3) = 10,000$?

$x = 997$ (D) Signup and view all the answers

Solve for $x$: $2 \cdot e^x = 16$

$x = \ln 8$ (A) Signup and view all the answers

Determine the value of $x$ in the equation, $10 \cdot e^{3x} = 250$.

$x = \frac{\ln{25}}{3}$ (C) Signup and view all the answers

Which of the following represents a continuous growth rate per year?

$q(t) = 50 \cdot e^{0.0168t}$ (B) Signup and view all the answers

A bacterial population is modeled by $f(t) = 50 \cdot e^{0.25t}$, where $t$ is in hours. Approximately how many bacteria are present after 10 hours?

608 (D) Signup and view all the answers

Using the bacteria model $f(t) = 50 \cdot e^{0.25t}$, approximately how long does it take for the initial population to double?

2.77 hours (B) Signup and view all the answers

The population of the U.S. is modeled by $p(t) = 192 \cdot e^{0.014t}$ million, where $t$ is years after 1964. What population does the model predict for 1974?

220.5 million (D) Signup and view all the answers

Using the population model $p(t) = 192 \cdot e^{0.014t}$, in which year does the model predict the population will reach 300 million?

2004 (A) Signup and view all the answers

Which function should only be used when change happens continuously?

$w(t) = 10 \cdot e^{0.005t}$ (D) Signup and view all the answers

Two population models for a city are given as $a(t) = 8.75 \cdot (1 + 0.01)^t$ and $b(t) = 8.75 \cdot e^{0.01t}$, where $t$ is years after 2010. What best describes model $a(t)$?

Discrete annual growth of approximately 1%. (A) Signup and view all the answers

For the bacterial growth model $b(t) = 50 \cdot e^{(0.25t)}$, what does the value 50 represent?

The initial bacteria population at time $t = 0$. (C) Signup and view all the answers

In the bacterial growth model $b(t) = 50 \cdot e^{(0.25t)}$, what is the continuous growth rate per hour, expressed as a percentage?

25% (B) Signup and view all the answers

Which of the following represents the missing information in the population model $p(t) = $______ $\cdot e^{t}$, given a continuous growth rate of 1.4% per year and an initial population of approximately 192 million in 1964?

$p(t) = 192 \cdot e^{0.014t}$ (D) Signup and view all the answers

What is the difference between representing exponential growth using a base of 'e' versus using a base of a decimal?

Using a base of 'e' allows for direct interpretation of the continuous growth rate, while a decimal base shows the overall growth factor per time period. (C) Signup and view all the answers

Which statement best describes the relationship between the function $f(t) = 10 \cdot (1.05)^t$ and $g(t) = 10 \cdot e^{(0.05t)}$?

Both functions represent exponential growth, and their graphs will appear very similar over a small interval, with $f(t)$ growing slightly faster. (A) Signup and view all the answers

Arrange the following exponential functions in order of increasing growth rate: $p(t) = 10 \cdot (1.01)^t$, $k(t) = 10 \cdot (1.03)^t$, $w(t) = 10 \cdot e^{(0.005t)}$, $q(t) = 10 \cdot e^{(0.01t)}$

$w(t), p(t), q(t), k(t)$ (A) Signup and view all the answers

If two bacterial colonies start with the same initial population and one grows according to the model $f(t) = 10 \cdot (1.05)^t$ while the other grows according to $g(t) = 10 \cdot e^{(0.05t)}$, what will be the approximate relationship between their populations after a significant amount of time?

The population of colony $f(t)$ will be slightly greater than the population of colony $g(t)$. (D) Signup and view all the answers

A scientist observes that a bacterial population doubles every 3 hours. Which model would best represent this growth, assuming an initial population of 20 bacteria?

$b(t) = 20 \cdot (2)^{(t/3)}$ (C) Signup and view all the answers

Flashcards

Mathematical constant e

A constant approximately equal to 2.718, used in exponential functions.

Exponential function

A function in the form f(t) = a * e^(bt), showing rapid growth changes.