How do you provide an opportunity for your students to figure out the relationship between differentiation and antidifferentiation?

We have used the Calculus Nspired activities The First Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus for several years now to improve our understanding of the relationship between a function and its accumulation function. I actually do print the student handouts for these activities and give students time during class to make sense of the relationship between differentiation and antidifferentiation.

It was time for our whole class discussion.

We defined the accumulation function using a definite integral. What do we know?

Students had figured out earlier that the definite integral of f(x) from a to a would be 0 and concluded that F(a)=0.

Students recognized that the value of the definite integral of f(x) from a to d would be F(d) and that the value of the definite integral of f(x) from a to c would be F(c).

Suppose we want to calculate the definite integral of f(x) from c to d. They could tell area-wise that was equivalent to finding the definite integral of f(x) from a to d and subtracting the definite integral of f(x) from a to c, which of course gives us F(d)-F(c).

And what does F(x) have to do with f(x)?

They could tell from the exploration that F(x) is the antiderivative of f(x).

Really? You mean we don’t have to do the limit-sum-infinite-number-of-rectangles every time? Really. You’ve earned the Fundamental Theorem of Calculus.

We talked for a little while about average value using the Calculus Nspired activity MVT for Integrals, and then checked in on their understanding.

As we moved into the second part of the Fundamental Theorem of Calculus, I posed a question to see how they would answer. (Remember that at this point, they’ve been using the FTOC for about 20 minutes.)

I was excited about a few students getting it right. Without discussing the correct responses with the whole group (I showed their answers but had Show Correct Answer deselected, I sent another question, which unearthed their misconception and revealed my initially bad question.

The students who got the answer correct in the first question had gotten it correct the wrong way, but their mistake wasn’t revealed because sin(-π)=0.

By now we were past the bell, and so we started over the next lesson with the second part of the Fundamental Theorem of Calculus.

It’s always exciting to find both the right questions to ask (the ones that reveal student misconceptions) and the wrong questions to ask (the ones that hide student misconceptions) so that I can continue asking the right ones and discontinue asking the wrong ones. In this lesson, I found at least one of each. And so the journey finding and asking the right questions continues …

howardat58

February 27, 2015 at 7:40 pm

So things change. When I was at school there was The Fundamental Theorem of Calculus. Now, according to TI there are two of them, and when I looked at the TI links neither of them was stated. Clearly a case of inflation of fundamentals.

jwilson828

February 27, 2015 at 8:51 pm

I was taught and have taught out of the Larson/Hostetler/Edwards Calculus textbook, which gives two FTOCs. The first is to calculate a definite integral, and the second shows the relationship between differentiation and antidifferentiation. In other texts, I’ve seen what Larson calls the second as the first, and what he calls the first as the second. In still other texts I’ve seen it listed as one FTOC with two parts.

howardat58

February 28, 2015 at 7:57 pm

The calculus jargon does change. We had indefinite integrals, then primitives, that was in the UK.

The funny thing about FTOC is that if you see the x variable as time and the f(t) as a flow rate and the definite integral as a function of its upper limit then the whole business is “almost obvious”. Which is what you were doing with the TI. Do you go all the way to the formal differentiation of the integral?

Ellen Browne

February 28, 2015 at 12:56 pm

Nice Jennifer! Sending this one to my calculus teaching husband.