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Questions and Answers
Which statement accurately distinguishes between exponential and linear growth?
Which statement accurately distinguishes between exponential and linear growth?
- Exponential growth involves multiplying by a constant factor over equal intervals, while linear growth involves adding a constant amount. (correct)
- Exponential growth applies only to populations, while linear growth applies only to financial investments.
- Linear growth eventually surpasses exponential growth due to its additive nature.
- Exponential growth involves adding a constant amount over equal intervals, while linear growth involves multiplying by a constant factor.
In the general form of an exponential function, $f(x) = ab^x$, what constraints apply to the base, $b$?
In the general form of an exponential function, $f(x) = ab^x$, what constraints apply to the base, $b$?
- $b$ must be an integer.
- $b$ must be greater than 1.
- $b$ can be any real number.
- $b$ must be a positive real number not equal to 1. (correct)
Consider two exponential decay functions, $f(x) = a(0.5)^x$ and $g(x) = a(0.75)^x$, with $a > 0$. Which statement accurately compares their rates of decay?
Consider two exponential decay functions, $f(x) = a(0.5)^x$ and $g(x) = a(0.75)^x$, with $a > 0$. Which statement accurately compares their rates of decay?
- Both functions decay at the same rate because they are both exponential functions.
- $f(x)$ decays faster than $g(x)$ because its base is smaller. (correct)
- The rates of decay cannot be compared without knowing the value of $a$.
- $g(x)$ decays faster than $f(x)$ because its base is larger.
Given an exponential function $f(x) = a \cdot b^x$ where $0 < b < 1$, which of the following transformations would result in an exponential growth function?
Given an exponential function $f(x) = a \cdot b^x$ where $0 < b < 1$, which of the following transformations would result in an exponential growth function?
How does continuous compounding differ fundamentally from compounding a finite number of times per year?
How does continuous compounding differ fundamentally from compounding a finite number of times per year?
An investment grows continuously at a nominal rate $r$. Which formula correctly calculates the percentage of interest earned relative to the principal at time $t$?
An investment grows continuously at a nominal rate $r$. Which formula correctly calculates the percentage of interest earned relative to the principal at time $t$?
Given two exponential functions, $f(x) = a \cdot e^{kx}$ and $g(x) = a \cdot e^{-kx}$, where $a > 0$ and $k > 0$, describe the relationship between these functions.
Given two exponential functions, $f(x) = a \cdot e^{kx}$ and $g(x) = a \cdot e^{-kx}$, where $a > 0$ and $k > 0$, describe the relationship between these functions.
If a radioactive substance decays exponentially, and its half-life is known, what information is needed to determine the remaining amount of the substance after a given time?
If a radioactive substance decays exponentially, and its half-life is known, what information is needed to determine the remaining amount of the substance after a given time?
Consider an investment that compounds continuously. How does the effective annual yield compare to the nominal APR?
Consider an investment that compounds continuously. How does the effective annual yield compare to the nominal APR?
In the context of exponential growth or decay, what is the significance of the natural logarithm (ln) in solving for time or rate variables?
In the context of exponential growth or decay, what is the significance of the natural logarithm (ln) in solving for time or rate variables?
Flashcards
Percent Change
Percent Change
A change based on a percentage of the original amount.
Exponential Growth
Exponential Growth
Increase at a constant multiplicative rate of change over equal time increments.
Exponential Decay
Exponential Decay
Decrease at a constant multiplicative rate of change over equal time increments.
Exponential Growth (value)
Exponential Growth (value)
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Linear Growth (value)
Linear Growth (value)
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Exponential Function (General Form)
Exponential Function (General Form)
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Exponential Growth Function
Exponential Growth Function
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Exponential Decay Function
Exponential Decay Function
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Nominal Interest Rate
Nominal Interest Rate
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Compound Interest Formula
Compound Interest Formula
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Study Notes
- Exponential functions model rapid growth, such as population increases.
Identifying Exponential Functions
- Linear growth involves a constant rate of change, where the output increases by a fixed number for each unit increase in input (e.g., f(x) = 3x + 4).
- Exponential relationships involve a percent change per unit time instead of a constant change.
- Percent change is a change based on a percentage of the original amount.
- Exponential growth is an increase based on a constant multiplicative rate of change over equal time increments, reflecting a percent increase of the original amount over time.
- Exponential decay is a decrease based on a constant multiplicative rate of change over equal time increments, reflecting a percent decrease of the original amount over time.
- Exponential growth dwarfs linear growth because the original value increases by the same percentage over equal increments.
- Linear growth sees the original value increase by the same amount over equal increments.
Defining an Exponential Function
- The general form of an exponential function is f(x) = abˣ, where a is a nonzero number and b is a positive real number not equal to 1.
- If b > 1, the function grows at a rate proportional to its size.
- If 0 < b < 1, the function decays at a rate proportional to its size.
- The domain of f is all real numbers.
- The range of f is all positive real numbers if a > 0.
- The range of f is all negative real numbers if a < 0.
- The y-intercept is (0, a), and the horizontal asymptote is y = 0.
Evaluating Exponential Functions
- To evaluate an exponential function, substitute the given x-value into the function and simplify.
- Standard form is f(x) = abˣ.
- Shifted form is f(x) = ab^(x-c) + d.
Finding the Equation of an Exponential Function
- Given two data points, create two equations by substituting the points into the form f(x) = abˣ.
- Solve the resulting system of two equations in two unknowns (a and b) to find the equation of the exponential function.
Understanding Compound Interest
- Principal (P) is the initial investment.
- Annual interest rate (r) is a percentage of the principal earned per year.
- Term (t) is the length of time over which interest is earned.
- Compounding refers to the process of earning interest on prior interest earned, along with the principal.
- Compound interest: A = P(1 + r/n)^(nt), where:
- A = value of the investment after t years.
- t = number of years.
- P = principal.
- r = annual interest rate.
- n = number of times the interest is compounded per year.
- Nominal rate is the stated interest rate, while effective rate is the actual interest rate earned after compounding.
Evaluating Exponential Functions with Base e
- e is an irrational number approximately equal to 2.7182818284.
- The function f(x) = eˣ is called the natural exponential function.
- Continuous growth/decay formula: f(x) = ae^(rt), where:
- a is the initial value.
- r is the continuous growth rate per unit of time.
- t is the elapsed time.
- If r > 0, the formula represents continuous growth.
- If r < 0, the formula represents continuous decay.
- An exponential decay function can be written as f(x) = a(e)^(-nx) for some positive number n.
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