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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 are the differences between local and global extrema?
Local extrema: max/min in a neighborhood. Global extrema: max/min over the entire domain.
What are the differences between the First and Second Derivative Tests?
First Derivative Test: uses sign changes of $f'(x)$ to find extrema. Second Derivative Test: uses the sign of $f''(x)$ at critical points to find extrema.
What are the differences between even and odd functions?
Even functions: symmetric about the y-axis, $f(-x) = f(x)$. Odd functions: symmetric about the origin, $f(-x) = -f(x)$.
Compare increasing vs. concave up.
Increasing: $f'(x) > 0$. Concave Up: $f''(x) > 0$. A function can be increasing and concave down, or increasing and concave up.
Compare decreasing vs. concave down.
Decreasing: $f'(x) < 0$. Concave Down: $f''(x) < 0$. A function can be decreasing and concave up, or decreasing and concave down.
What is the difference between a critical point and a point of inflection?
Critical Point: $f'(x) = 0$ or undefined (potential max/min). Point of Inflection: $f''(x) = 0$ or undefined (concavity change).
What is the difference between a root and a y-intercept?
Root: where the function crosses the x-axis. Y-intercept: where the function crosses the y-axis.
Compare finding critical points of $f(x)$ vs. $f'(x)$.
Critical points of $f(x)$: $f'(x) = 0$ or undefined. Critical points of $f'(x)$: $f''(x) = 0$ or undefined.
Compare even symmetry vs. odd symmetry.
Even symmetry: symmetric about the y-axis. Odd symmetry: symmetric about the origin.
What is the difference between finding where a function is zero versus undefined?
Zero: Find where the function equals zero. Undefined: Find where the function has a discontinuity (e.g., division by zero).
Explain the relationship between the first derivative and increasing/decreasing intervals.
If $f'(x) > 0$, $f(x)$ is increasing. If $f'(x) < 0$, $f(x)$ is decreasing.
Explain the First Derivative Test.
If $f'(x)$ changes from positive to negative at $x=c$, then $f(x)$ has a local maximum at $x=c$. If $f'(x)$ changes from negative to positive at $x=c$, then $f(x)$ has a local minimum at $x=c$.
Explain the relationship between the second derivative and concavity.
If $f''(x) > 0$, $f(x)$ is concave up. If $f''(x) < 0$, $f(x)$ is concave down.
Explain how to find points of inflection.
Find where $f''(x) = 0$ or is undefined, and check if the concavity changes at those points.
How do you determine symmetry about the y-axis?
Check if $f(-x) = f(x)$. If true, the function is even and symmetric about the y-axis.
How do you determine symmetry about the origin?
Check if $f(-x) = -f(x)$. If true, the function is odd and symmetric about the origin.
What does a discontinuity in a function indicate?
A point where the function is not continuous, possibly indicating a hole, jump, or vertical asymptote.
What is the significance of critical points?
Critical points are potential locations of local maxima, local minima, or saddle points of a function.
Explain the Second Derivative Test.
Uses the second derivative to determine if a critical point is a local max or min. If $f''(c) > 0$, local minimum. If $f''(c) < 0$, local maximum.
What is the domain of a polynomial function?
All real numbers, unless otherwise restricted.