All Flashcards
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.
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.