All Flashcards
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 is the formula for the average rate of change of a function f(x) over the interval [a, b]?
What is the condition stated by the Mean Value Theorem?
for some
How do you find critical points?
Solve or find where is undefined.
How do you determine concavity using the second derivative?
implies concave up; implies concave down.
How do you find inflection points?
Solve or find where is undefined, and verify concavity changes.
What does f'(x) > 0 imply?
f(x) is increasing.
What does f'(x) < 0 imply?
f(x) is decreasing.
What does f''(x) = 0 imply?
Possible inflection point.
What does the second derivative test tell us about local extrema?
If and , then is a local minimum. If and , then is a local maximum.
What is the general approach to solving optimization problems?
- Define the objective function. 2. Identify constraints. 3. Find critical points. 4. Test for extrema.
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.