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Explain how the sign of the first derivative relates to the function's behavior.
Positive derivative: increasing function. Negative derivative: decreasing function. Zero derivative: potential extremum or inflection point.
How does the First Derivative Test help find local extrema?
By identifying critical points and analyzing the sign change of the derivative around those points.
Explain why a critical point doesn't always indicate a local extremum.
The derivative might not change sign at the critical point (e.g., a saddle point).
What does the First Derivative Test tell you about the graph of (f(x))?
It helps identify where the graph has peaks (local maxima) and valleys (local minima), indicating changes in the function's direction.
How do you determine if a critical point is a local max using the first derivative test?
If (f'(x)) changes from positive to negative at the critical point, it's a local maximum.
How do you determine if a critical point is a local min using the first derivative test?
If (f'(x)) changes from negative to positive at the critical point, it's a local minimum.
What happens if the first derivative does not change sign at a critical point?
The critical point is neither a local maximum nor a local minimum; it could be a saddle point or an inflection point.
Why is it important to find all critical points when using the First Derivative Test?
To ensure that all potential locations of local extrema are considered, as extrema can only occur at critical points or endpoints.
Explain the limitations of the First Derivative Test.
The First Derivative Test can only identify local extrema and doesn't provide information about global extrema or concavity.
What is the relationship between the sign of the derivative and the slope of the tangent line?
The sign of the derivative at a point indicates whether the tangent line at that point has a positive (increasing), negative (decreasing), or zero (horizontal) slope.
How to find relative extrema using the First Derivative Test?
1. Find critical points. 2. Determine the sign of the derivative on either side of each critical point. 3. Apply the First Derivative Test rules.
Steps to find critical points of a function.
1. Find the derivative of the function. 2. Set the derivative equal to zero and solve for x. 3. Find where the derivative is undefined.
How to determine if a function is increasing on an interval?
1. Find the derivative. 2. Determine where the derivative is positive.
How to determine if a function is decreasing on an interval?
1. Find the derivative. 2. Determine where the derivative is negative.
Steps to find the relative minima of (h(x)) given (h'(x) = x^2(x-3)(x+4)).
1. Set (h'(x) = 0) to find critical points: (x = -4, 0, 3). 2. Test intervals around each critical point to determine the sign change of (h'(x)). 3. Identify where (h'(x)) changes from negative to positive.
Steps to find the relative maxima of (g(x) = x^5 - 80x).
1. Find (g'(x) = 5x^4 - 80). 2. Set (g'(x) = 0) to find critical points: (x = -2, 2). 3. Test intervals around each critical point to determine the sign change of (g'(x)). 4. Identify where (g'(x)) changes from positive to negative.
How do you test the intervals around a critical point?
Choose a test value within each interval and evaluate the derivative at that value. The sign of the derivative indicates whether the function is increasing or decreasing in that interval.
What should you do if a critical point is an endpoint of the domain?
Evaluate the derivative only on the side of the critical point that is within the domain. The sign of the derivative on that side determines if the endpoint is a local extremum.
How do you find the y-coordinate of a local extremum?
Substitute the x-coordinate of the local extremum into the original function (f(x)) to find the corresponding y-coordinate.
How do you handle critical points where the derivative is undefined?
Treat them similarly to where the derivative is zero: test the intervals around the critical point to determine the sign change of the derivative.
What does a horizontal tangent line on the graph of (f(x)) indicate?
A critical point where (f'(x) = 0), which could be a local maximum, local minimum, or saddle 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 (local max) or decreasing to increasing (local min).
How does the graph of (f'(x)) relate to the increasing/decreasing behavior of (f(x))?
Where (f'(x) > 0), (f(x)) is increasing. Where (f'(x) < 0), (f(x)) is decreasing.
What does the x-intercept of the graph of (f'(x)) represent?
It represents a critical point of the original function (f(x)).
How can you determine if a critical point is a local maximum from the graph of (f'(x))?
If (f'(x)) changes from positive to negative at the x-intercept, the corresponding point on (f(x)) is a local maximum.
How can you determine if a critical point is a local minimum from the graph of (f'(x))?
If (f'(x)) changes from negative to positive at the x-intercept, the corresponding point on (f(x)) is a local minimum.
What does it mean if (f'(x)) is always positive?
\(f(x)) is always increasing.
What does it mean if (f'(x)) is always negative?
\(f(x)) is always decreasing.
How can you identify intervals where (f(x)) is increasing or decreasing by looking at the graph of (f'(x))?
If (f'(x)) is above the x-axis, (f(x)) is increasing. If (f'(x)) is below the x-axis, (f(x)) is decreasing.
How does the steepness of the graph of (f(x)) relate to the value of (f'(x))?
The steeper the graph of (f(x)), the larger the absolute value of (f'(x)).