Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.
I can find antiderivatives with knowledge of derivatives.
I can explain why a constant of integration is needed when evaluating an indefinite integral.
Quick Lesson Plan
The focus today is on graphical representations of functions and their antiderivatives. Using decks of 24 cards with 12 original functions [f(x)] and 12 matching antiderivatives [F(x)], students use a visual approach to create matched pairs: a given integrand, f(x), and the correct antiderivative, F(x).
Activity question 1 presents a simple question whose answer can be as complex as time allows. After discussing which graph may represent the derivative function, you may wish to use this question as an excellent opportunity to review of the concepts surrounding increasing/decreasing functions, relative maxima, and concavity.
“A Match Made in Heaven” reverses the process and directs students to identify antiderivatives (some familiar and some unfamiliar). Three of the cards have a graph of the same original function: f(x) = -0.5. Students should identify three different, yet correct, antiderivative graphs: F(x) = -0.5x – 1, F(x) = -0.5x – 4, and F(x) = -0.5x + 3. Could a “family” of antiderivatives suffice? Perhaps F(x) = -0.5x + C? This activity leads students, of course, to the need for a constant of integration when working with indefinite integrals.
Inclusion of the “constant of integration” is an important component of a student’s work with indefinite integrals. For specific examples of how the constants are scored in FRQs, please see 2013 AB 6b, 2014 AB 6c, 2016 AB 4c, or 2017 AB 4c.
The Important Ideas section of today’s lesson includes several rules and integration formulas which students should memorize immediately. Place special focus on integrands leading to natural log expressions (they’re often forgotten), the need for absolute value in logarithm expressions (why is absolute value required?), and the placement of negative signs when integrating sin(x).