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What are the differences between exponential growth and exponential decay?
Growth: b > 1, increases rapidly | Decay: 0 < b < 1, decreases rapidly.
What are the differences between linear and exponential functions?
Linear: Constant rate of change | Exponential: Constant proportional rate of change.
What are the differences between using 'e' and another base 'b' in exponential models?
e: Models continuous growth/decay, related to natural logarithm | b: Models growth/decay over discrete intervals.
What are the differences between $f(x) = e^x$ and $f(x) = e^{-x}$?
$e^x$: Exponential growth | $e^{-x}$: Exponential decay.
Compare and contrast exponential and polynomial functions.
Exponential: Variable in exponent, rapid growth/decay | Polynomial: Variable in base, growth depends on degree.
Compare and contrast exponential growth and logistic growth.
Exponential: Unrestricted growth | Logistic: Growth limited by carrying capacity.
Compare and contrast exponential functions with positive and negative exponents.
Positive: Increasing function | Negative: Decreasing function.
Compare and contrast exponential functions with different initial values.
Different y-intercepts, but same growth/decay rate if 'b' is the same.
Compare and contrast exponential regression and linear regression.
Exponential: Fits exponential model | Linear: Fits linear model. Use $R^2$ to determine best fit.
Compare and contrast the effects of adding a constant inside vs. outside the exponential function.
Inside: $f(x) = a b^{(x+c)}$: Horizontal shift | Outside: $f(x) = ab^x + c$: Vertical shift.
Explain how adding a constant to the dependent variable can reveal an exponential relationship.
Shifts the y-intercept, allowing proportional growth to be observed from a non-zero baseline.
Explain the significance of the base 'b' in an exponential function.
Represents the constant proportion by which the output value is multiplied at each step.
Why is the natural base 'e' used in exponential functions?
Models continuous growth/decay and is the base of the natural logarithm.
Explain how equivalent forms of exponential functions can provide different interpretations.
Changing the base or exponent can reveal the growth rate over different time intervals.
Describe how to construct an exponential function using two input-output pairs.
Set up a system of equations and solve for 'a' (initial value) and 'b' (base).
Explain how exponential regression works.
Fits an exponential function to a data set by minimizing the difference between observed and predicted values.
What does $R^2$ close to 1 imply?
The model is a good fit for the data.
How does the value of 'b' affect the graph of $f(x) = ab^x$?
If b > 1, the graph increases exponentially. If 0 < b < 1, the graph decreases exponentially.
Why is 'e' important in calculus?
It simplifies many calculations, especially those involving derivatives and integrals of exponential functions.
What are the key characteristics of exponential functions?
Constant ratio between successive y-values for equally spaced x-values, horizontal asymptote, no x-intercept (unless a constant is added).
Define exponential growth.
Growth increasing at an increasing rate; b > 1 in $f(x) = ab^x$.
Define exponential decay.
Decay decreasing at a decreasing rate; 0 < b < 1 in $f(x) = ab^x$.
What is the general form of an exponential function?
$f(x) = ab^x$, where 'a' is the initial value and 'b' is the base.
Define the natural base 'e'.
A mathematical constant approximately equal to 2.71828, used for continuous growth/decay.
What is exponential regression?
A method to fit an exponential function to a set of data points.
Define the correlation coefficient $R^2$ in exponential regression.
A measure of how well the exponential model fits the data.
What are residuals in exponential regression?
The difference between the observed data and the predicted values from the exponential model.
What is 'ln'?
The natural logarithm, the inverse function of $e^x$.
What is an equivalent form of an exponential function?
A different representation of the same function that reveals different properties.
What does the initial value 'a' represent in $f(x) = ab^x$?
The value of the function when x = 0.