Define 'derivative'.
The derivative of a function is the rate of change of the function at a given point.
What are 'critical points'?
Points where the function's derivative equals 0 or is undefined, and the points where the function itself is undefined.
What does it mean for a function to be 'increasing'?
A function is increasing on an interval if its values increase as the input increases.
What does it mean for a function to be 'decreasing'?
A function is decreasing on an interval if its values decrease as the input increases.
Define 'rate of change'.
The rate at which a quantity is increasing or decreasing.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.
Define 'undefined' in the context of functions.
A value for which the function produces no output or an infinite output.
What is an 'interval'?
A set of real numbers between two specified values.
Define 'test point' in the context of finding increasing/decreasing intervals.
A value chosen within an interval to evaluate the derivative and determine the function's behavior on that interval.
What is the 'number line' used for in this context?
A visual representation of real numbers used to divide the domain into intervals based on critical points.
How does a positive derivative relate to a function's behavior?
If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.
How does a negative derivative relate to a function's behavior?
If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.
Why are critical points important for determining increasing/decreasing intervals?
A function can only change from increasing to decreasing (or vice versa) at critical points.
Explain the process of using test points to determine function behavior.
Choose a test point in each interval defined by critical points, evaluate $f'(x)$ at that point. The sign of $f'(x)$ indicates whether $f(x)$ is increasing or decreasing on that interval.
What does $f'(x) = 0$ indicate about the function $f(x)$?
It indicates a critical point where the function may have a local maximum, local minimum, or a point of inflection.
How do you determine if a function is increasing or decreasing at a specific point?
Evaluate the derivative of the function at that point. If the derivative is positive, the function is increasing; if negative, it's decreasing.
Explain the relationship between the sign of the derivative and the slope of the tangent line.
A positive derivative means the tangent line has a positive slope (increasing function), and a negative derivative means the tangent line has a negative slope (decreasing function).
Why do we need to consider points where the function is undefined?
Because the function's behavior can change at these points, even though they are not critical points in the traditional sense.
How does the first derivative test help in identifying local extrema?
The first derivative test uses the sign changes of the first derivative around a critical point to determine if it's a local maximum or minimum.
Explain how to use the sign chart of the first derivative to determine intervals of increasing and decreasing behavior.
Create a sign chart with critical points, test values, and the sign of the derivative. Positive signs indicate increasing intervals, and negative signs indicate decreasing intervals.
What is the power rule for derivatives?
$\frac{d}{dx}(x^n) = nx^{n-1}$
What is the derivative of a constant?
$\frac{d}{dx}(c) = 0$
What is the sum rule for derivatives?
$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
What is the constant multiple rule for derivatives?
$\frac{d}{dx}[cf(x)] = cf'(x)$
What is the product rule for derivatives?
$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
What is the quotient rule for derivatives?
$\frac{d}{dx}[ rac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
What is the chain rule for derivatives?
$\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$
What is the derivative of $\sin(x)$?
$\frac{d}{dx}(\sin(x)) = \cos(x)$
What is the derivative of $\cos(x)$?
$\frac{d}{dx}(\cos(x)) = -\sin(x)$
What is the derivative of $e^x$?
$\frac{d}{dx}(e^x) = e^x$
