All Flashcards
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 Mean Value Theorem guarantee?
If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that .
What does the Extreme Value Theorem guarantee?
If f(x) is continuous on [a, b], then f(x) attains both a maximum and a minimum value on that interval.
What is the application of the Mean Value Theorem?
It is used to relate the average rate of change of a function over an interval to its instantaneous rate of change at some point within the interval.
What is the application of the Extreme Value Theorem?
It guarantees the existence of absolute maximum and minimum values for continuous functions on closed intervals, which is crucial for optimization problems.
What are the conditions for the Mean Value Theorem to apply?
Function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
What are the conditions for the Extreme Value Theorem to apply?
Function must be continuous on the closed interval [a, b].
How does the Mean Value Theorem relate to Rolle's Theorem?
Rolle's Theorem is a special case of the Mean Value Theorem where f(a) = f(b).
How is the Extreme Value Theorem used in optimization?
It ensures that a continuous function on a closed interval has a maximum and minimum value, allowing us to find the optimal solution.
What does the Mean Value Theorem tell us about the relationship between a function and its derivative?
It states that at some point in an interval, the derivative of the function is equal to the average rate of change over that interval.
What does the Extreme Value Theorem guarantee about the existence of extrema?
It guarantees that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum within that interval.
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.