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.
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.
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.