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What are the differences between local and global extrema?
Local: Extrema within a specific interval. Global: Extrema over the entire domain.
What are the differences between the first derivative test and the second derivative test?
First Derivative: Uses the sign of f'(x) to determine increasing/decreasing and local extrema. Second Derivative: Uses the sign of f''(x) to determine concavity and local extrema.
What are the differences between concave up and concave down?
Concave Up: f''(x) > 0, curve opens upwards. Concave Down: f''(x) < 0, curve opens downwards.
What are the differences between critical points and inflection points?
Critical Points: f'(x) = 0 or undefined, potential local extrema. Inflection Points: f''(x) changes sign, change in concavity.
What are the differences between minimization and maximization problems?
Minimization: Finding the minimum value of a function. Maximization: Finding the maximum value of a function.
What are the differences between the graphical and analytical methods for solving optimization problems?
Graphical: Sketching the graph to find extrema. Analytical: Using calculus (derivatives) to find extrema.
What are the differences between using f'(x) and f''(x) when sketching a graph?
f'(x): Determines increasing/decreasing intervals and local extrema. f''(x): Determines concavity and inflection points.
What are the differences between the Mean Value Theorem and the Extreme Value Theorem?
MVT: Guarantees a point where the instantaneous rate of change equals the average rate of change. EVT: Guarantees the existence of a maximum and minimum value on a closed interval.
What are the differences between relative and absolute extrema?
Relative: Local maximum or minimum within a specific interval. Absolute: Global maximum or minimum over the entire domain.
What are the differences between a function and its derivative?
Function: Represents the original relationship between x and y. Derivative: Represents the rate of change of the function.
What does the x-intercept of f'(x) tell you about f(x)?
It indicates a critical point of f(x), where f(x) may have a local max or min.
What does the sign of f'(x) tell you about the graph of f(x)?
Positive f'(x) means f(x) is increasing; negative f'(x) means f(x) is decreasing.
What does the sign of f''(x) tell you about the graph of f(x)?
Positive f''(x) means f(x) is concave up; negative f''(x) means f(x) is concave down.
How can you identify inflection points from the graph of f''(x)?
Inflection points occur where f''(x) changes sign (crosses the x-axis).
If f'(x) is always positive, what does that imply about f(x)?
f(x) is always increasing.
If f''(x) is always negative, what does that imply about f(x)?
f(x) is always concave down.
What does a horizontal tangent line on the graph of f(x) indicate?
It indicates that f'(x) = 0 at that point, a critical point.
How can you identify local extrema on the graph of f(x)?
Look for points where the graph changes direction (from increasing to decreasing or vice versa).
What does the area under the curve of f'(x) represent?
The net change in f(x) over the given interval.
How can you determine where f(x) has a local maximum from the graph of f'(x)?
Look for where f'(x) changes from positive to negative.
Define the Mean Value Theorem (MVT).
If a function is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
Define the Extreme Value Theorem (EVT).
If a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a maximum and a minimum value on that interval.
What is a critical point of a function f(x)?
A value c in the domain of f(x) such that either f'(c) = 0 or f'(c) does not exist.
Define global (absolute) extrema.
The highest and lowest points of a function over its entire domain.
Define local (relative) extrema.
The highest and lowest points of a function over a specific subinterval of its domain.
Define concavity.
The curvature of a function at a given point; indicates whether the function is 'bending up' or 'bending down'.
Define an inflection point.
A point on a curve where the concavity changes.
Define optimization problems.
Mathematical problems that involve finding the best solution (minimum or maximum) among a set of possible solutions.
Define the first derivative test.
A method used to determine whether a function is increasing or decreasing on a specific interval by analyzing the sign of its first derivative.
Define the Candidates Test.
A method used to determine the absolute extrema of a continuous function on a closed interval by evaluating the function at critical points and endpoints.