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Steps to find intervals where $f(x)$ is increasing/decreasing.
1. Find $f'(x)$. 2. Find critical points (where $f'(x) = 0$ or is undefined). 3. Create intervals using critical points. 4. Choose test points in each interval. 5. Evaluate $f'(x)$ at each test point. 6. Determine if $f(x)$ is increasing ($f'(x) > 0$) or decreasing ($f'(x) < 0$).
How to find critical points?
1. Find the derivative $f'(x)$. 2. Set $f'(x) = 0$ and solve for $x$. 3. Identify $x$ values where $f'(x)$ is undefined.
How to choose appropriate test points?
Select any value within each interval created by the critical points. The test point should be easy to evaluate in the derivative.
How do you handle a function with a discontinuity when finding increasing/decreasing intervals?
Include the point of discontinuity when creating intervals, as the function's behavior can change there.
What do you do if the derivative is always positive?
The function is always increasing on its domain.
What do you do if the derivative is always negative?
The function is always decreasing on its domain.
How do you determine the intervals of increasing and decreasing behavior for a piecewise function?
Find the derivative of each piece of the function, determine critical points within each piece, and analyze the sign of the derivative in each interval, considering the points where the pieces connect.
How do you determine the intervals of increasing and decreasing behavior for an implicit function?
Use implicit differentiation to find $\frac{dy}{dx}$, set $\frac{dy}{dx} = 0$ to find critical points, and analyze the sign of $\frac{dy}{dx}$ in each interval to determine increasing and decreasing behavior.
How do you handle absolute value functions when finding increasing/decreasing intervals?
Rewrite the absolute value function as a piecewise function, find the derivative of each piece, determine critical points, and analyze the sign of the derivative in each interval.
How do you determine the intervals of increasing and decreasing behavior for trigonometric functions?
Find the derivative, set it equal to zero to find critical points, and analyze the sign of the derivative in each interval, considering the periodic nature of the trigonometric functions.
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 $e^x$?
$\frac{d}{dx}(e^x) = e^x$
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.