AP Calculus AB: Lesson 6.6 Fundamental Theorem of Calculus, Part 2

In this segment, Michelle Crummel introduces the second part of the Fundamental Theorem of Calculus. She explains the concept of antiderivatives and demonstrates how to find the antiderivative of a given function, using the reverse power rule. Michelle then shows how to substitute variables to rewrite the antiderivative in terms of a different variable and proceeds to illustrate the process of taking the derivative of an antiderivative, which results in the original function. This is followed by examples using different functions, such as cosine of t and t cubed, to further clarify the theorem's application. Michelle emphasizes the relationship between the derivative of the antiderivative and the original function.

This paragraph delves into applying the second part of the Fundamental Theorem of Calculus to definite integrals. Michelle explains how to evaluate a definite integral using the antiderivative function and then demonstrates how taking the derivative of a definite integral results in the original function, with the constant term disappearing. She uses the chain rule to show how to handle definite integrals with variable limits of integration and simplifies the process using a shortcut. Michelle also explains how to deal with situations where the lower limit of integration is a constant, which simplifies the derivative calculation.

In this section, Michelle explores how to determine the intervals on which a function defined by a definite integral is increasing or decreasing. She explains that the function is increasing where its derivative is positive and decreasing where the derivative is negative. Michelle uses the graph of a function to illustrate this concept and provides a step-by-step guide on how to identify these intervals based on the function's values and the x-axis. She also touches on the concept of concavity, explaining how to determine where the function is concave up or down by examining the sign of the second derivative.

Michelle continues the lesson by showing how to calculate areas under a curve from a given graph and how to determine derivatives at specific points. She uses the graph of a function to find the definite integral's value at certain points, which corresponds to the area under the curve. She also explains how to find the first and second derivatives of a function at given values by using the fundamental theorem of calculus and the concept of the derivative. Michelle demonstrates this with examples, including reading values from the graph and applying the chain rule to find the second derivative.

This paragraph presents a series of comprehensive examples that apply the concepts of the fundamental theorem of calculus, including finding the antiderivative, evaluating definite integrals, and calculating derivatives. Michelle works through problems that involve various functions and intervals, showing how to compute the areas under the curve and how to find the function's values and derivatives at specific points. She also addresses the implications of integrating over intervals that proceed from right to left, which affects the sign of the calculated areas.

In the concluding segment, Michelle summarizes the key points covered in the lesson about the second part of the Fundamental Theorem of Calculus. She reiterates the process of finding antiderivatives, evaluating definite integrals, and calculating derivatives of these integrals. Michelle also highlights the importance of understanding how to work with variable limits of integration and the significance of the theorem in calculus. She wraps up the lesson by previewing the next topic, which is u-substitution, and thanks the viewers for joining the session.