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Formula for testing even function symmetry?
$f(-x) = f(x)$
Formula for testing odd function symmetry?
$f(-x) = -f(x)$
Second Derivative Test Formula
If $f''(c) > 0$, local minimum at $x=c$. If $f''(c) < 0$, local maximum at $x=c$.
How to find x-intercepts?
Set $f(x) = 0$ and solve for $x$.
How to find y-intercepts?
Set $x = 0$ and solve for $f(0)$.
What is the condition when $f$ is increasing?
$f'(x) > 0$
What is the condition when $f$ is decreasing?
$f'(x) < 0$
Condition for concave up?
$f''(x) > 0$
Condition for concave down?
$f''(x) < 0$
How do you find critical points?
Solve $f'(x) = 0$ or where $f'(x)$ is undefined.
Define critical point.
A point where the function's first derivative is zero or undefined.
What is a point of inflection?
A point where the concavity of a function changes.
Define local maximum.
A point where the function's value is greater than or equal to the values at all nearby points.
What is a local minimum?
A point where the function's value is less than or equal to the values at all nearby points.
Define concavity.
The direction in which a curve bends; either concave up or concave down.
What does symmetry mean for a function?
A function is symmetric if it looks the same when reflected across a line or point.
Define even function.
A function where $f(-x) = f(x)$ for all $x$ in the domain.
Define odd function.
A function where $f(-x) = -f(x)$ for all $x$ in the domain.
Define x-intercept.
The point(s) where the graph of a function intersects the x-axis.
Define y-intercept.
The point(s) where the graph of a function intersects the y-axis.
What does $f'(x) = 0$ mean on the graph of $f(x)$?
It indicates a horizontal tangent line, which could be a local maximum, local minimum, or saddle point.
What does $f'(x) > 0$ mean on the graph of $f(x)$?
The function $f(x)$ is increasing.
What does $f'(x) < 0$ mean on the graph of $f(x)$?
The function $f(x)$ is decreasing.
What does $f''(x) = 0$ mean on the graph of $f(x)$?
It indicates a possible point of inflection where the concavity may change.
What does $f''(x) > 0$ mean on the graph of $f(x)$?
The function $f(x)$ is concave up.
What does $f''(x) < 0$ mean on the graph of $f(x)$?
The function $f(x)$ is concave down.
How to identify local maxima on a graph?
Look for points where the function changes from increasing to decreasing.
How to identify local minima on a graph?
Look for points where the function changes from decreasing to increasing.
What does a sharp corner on the graph of $f(x)$ indicate about $f'(x)$?
It indicates that $f'(x)$ is undefined at that point.
How does the steepness of $f(x)$ relate to $f'(x)$?
The steeper the graph of $f(x)$, the larger the absolute value of $f'(x)$.