All Flashcards
Explain the Mean Value Theorem in simple terms.
At some point between two points on a curve, the instantaneous rate of change (derivative) equals the average rate of change (slope of the secant line).
Explain the Extreme Value Theorem in simple terms.
A continuous function on a closed interval is guaranteed to have a maximum and minimum value within that interval.
Explain how the first derivative test is used to find local extrema.
By analyzing the sign change of f'(x) around a critical point, we can determine if the point is a local max (f' changes from + to -) or local min (f' changes from - to +).
Explain how the second derivative test is used to determine concavity.
The sign of the second derivative indicates the concavity of the function. Positive means concave up, negative means concave down.
How are critical points related to local extrema?
Local extrema can only occur at critical points (where f'(x) = 0 or is undefined).
What are the conditions for applying the Mean Value Theorem?
The function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
What are the conditions for applying the Extreme Value Theorem?
The function must be continuous on the closed interval [a, b].
Explain how to find absolute extrema using the Candidates Test.
Evaluate the function at all critical points and endpoints within the interval. The largest value is the absolute maximum, and the smallest is the absolute minimum.
Explain the relationship between a function, its first derivative, and its second derivative.
f'(x) indicates where f(x) is increasing/decreasing. f''(x) indicates the concavity of f(x).
Explain the purpose of optimization problems.
To find the best possible solution (maximum or minimum) to a problem under given constraints.
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.