Topics 5.1-5.5 Review

Front

What are critical points in a function?

Back

Critical points are values of x where the derivative of the function is zero or undefined. They are important for determining local maxima and minima.

Front

How do you find critical points of a polynomial function?

Back

To find critical points, take the derivative of the polynomial, set it equal to zero, and solve for x.

Front

What does the Mean Value Theorem (MVT) state?

Back

The MVT states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the instantaneous rate of change at c equals the average rate of change over [a, b].

Front

How do you apply the Mean Value Theorem to find a point on a curve?

Back

Calculate the average rate of change between two points, then find where the derivative equals this rate to locate the point on the curve.

Front

What is the significance of a function's derivative being zero?

Back

When the derivative of a function is zero at a point, it indicates a potential local maximum, minimum, or a point of inflection.

Front

What does it mean for a function to be increasing or decreasing?

Back

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative.

Front

How can you determine where a function is decreasing using its derivative?

Back

Identify the intervals where the derivative is negative; these intervals indicate where the function is decreasing.

Front

What is the relationship between the number of zeros of a derivative and the behavior of the original function?

Back

The number of zeros of a derivative indicates potential local extrema in the original function, as each zero can correspond to a maximum or minimum.

Front

How do you find the average rate of change of a function between two points?

Back

The average rate of change is calculated as the difference in the function values at the two points divided by the difference in the x-values: (f(b) - f(a)) / (b - a).

Front

What is a differentiable function?

Back

A differentiable function is one that has a derivative at every point in its domain, meaning it is smooth and has no sharp corners or discontinuities.