Curve Sketching

Front

What is the Mean Value Theorem?

Back

The Mean Value Theorem 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 f'(c) = (f(b) - f(a)) / (b - a).

Front

How do you find points of inflection?

Back

Points of inflection occur where the second derivative of a function changes sign. To find them, set the second derivative equal to zero and solve for x.

Front

What does it mean for a function to be concave down?

Back

A function is concave down on an interval if its second derivative is negative on that interval. This means the graph of the function is curving downwards.

Front

What is Rolle's Theorem?

Back

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0.

Front

How do you determine intervals of concavity?

Back

To determine intervals of concavity, find the second derivative of the function, set it to zero to find critical points, and test intervals around those points to see where the second derivative is positive (concave up) or negative (concave down).

Front

What is the significance of critical points in curve sketching?

Back

Critical points, where the first derivative is zero or undefined, are significant because they indicate potential local maxima, minima, or points of inflection on the graph of the function.

Front

How do you verify the Mean Value Theorem for a function?

Back

To verify the Mean Value Theorem, check that the function is continuous on the closed interval and differentiable on the open interval, then find c such that f'(c) equals the average rate of change over the interval.

Front

What is the relationship between the first and second derivatives?

Back

The first derivative indicates the slope of the function (increasing or decreasing), while the second derivative indicates the concavity of the function (concave up or down).

Front

How can you identify local extrema using the first derivative test?

Back

To identify local extrema using the first derivative test, find critical points, determine the sign of the first derivative before and after each critical point, and conclude whether the function is increasing or decreasing.

Front

What is the role of the second derivative in curve sketching?

Back

The second derivative helps determine the concavity of the function and the location of points of inflection, which are crucial for accurately sketching the graph.