All Flashcards
How do you find intervals where f(x) is increasing or decreasing?
- Find f'(x). 2. Determine critical points where f'(x) = 0 or is undefined. 3. Test intervals between critical points using f'(x) to determine if it's positive (increasing) or negative (decreasing).
How do you find local extrema using the first derivative test?
- Find critical points. 2. Test the sign of f'(x) to the left and right of each critical point. 3. If f'(x) changes sign, a local extremum exists.
How do you determine the concavity of a function?
- Find f''(x). 2. Determine where f''(x) = 0 or is undefined. 3. Test intervals using f''(x) to determine if it's positive (concave up) or negative (concave down).
How do you find inflection points?
- Find f''(x). 2. Solve for f''(x) = 0 or where it is undefined. 3. Verify that the concavity changes at those points.
How do you find absolute extrema on a closed interval?
- Find critical points within the interval. 2. Evaluate f(x) at critical points and endpoints. 3. Compare values to find the absolute maximum and minimum.
How do you solve an optimization problem?
- Define the objective function. 2. Identify constraints. 3. Express the objective function in terms of one variable. 4. Find critical points. 5. Test for extrema.
How do you apply the Mean Value Theorem to a problem?
- Verify the function is continuous and differentiable on the given interval. 2. Calculate the average rate of change: (f(b)-f(a))/(b-a). 3. Find c such that f'(c) equals the average rate of change.
How do you use the second derivative test to find local extrema?
- Find critical points. 2. Calculate the second derivative, f''(x). 3. Evaluate f''(x) at each critical point. 4. If f''(x) > 0, local minimum. If f''(x) < 0, local maximum.
How do you determine if a critical point is a local max, min, or neither?
Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)).
How do you sketch a graph of f(x) given f'(x) and f''(x)?
- Identify critical points from f'(x). 2. Determine intervals of increasing/decreasing from f'(x). 3. Identify inflection points from f''(x). 4. Determine concavity from f''(x). 5. Sketch the graph using this information.
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 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.