How do you find intervals where f(x) is increasing or decreasing?
1. 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?
1. 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?
1. 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?
1. 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?
1. 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?
1. 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?
1. 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?
1. 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)?
1. 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.
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.
