I’m met with silence, which is expected. The girls in the front look attentive, batting their lashes, and the guys kind of slump back in their seats, hoping I won’t call on them.
“The linear approximation is the same as finding the tangent line to the curve,” I begin. “Which is what we’ve been doing all semester. Thankfully, this is a process we’re familiar with. But why would we want it?”
More silence. They need a little warmup.
“All right, let’s back up. Whatisthe tangent line?”
The whole class makes kissing sounds back at me.
“Right, right. It’s the line that kisses the graph of the function, barely touching it and then moving on. So why do we want to find that?”
Allison, a girl in the front row, raises her hand. “To find the tangent line to the curve?”
That’s what I just said.But I nod. “Yes. And why would we want that?”
“To find an approximation for the function?” Cruz, a veteran in the back corner, calls out.
“Exactly. Butwhy?”
I wait. Usually when I wait, the answer comes.
Kelly, a shy Asian girl in the middle, raises her hand. “Because it’s easier to use a linear function than the complicated functions.”
“Yes!” I point at her, and she blushes, and the girls in the front row scowl at her. I want to roll my eyes, but I don’t. “Imagine you’re on a desert island.”
“What book would you take with you?” James, the resident class clown, calls out to me from the back row.
“East of Eden.But that’s not the point. We’re not sitting on this desert island, reading books. We’re trying to make a raft to get off the island. And in order to build this raft, one critical part of the computation is the fifth root of thirty-three. Unfortunately, you didn’t bring a calculator. But you did bring your calculus knowledge.”
Blank stares look back at me.
“It’s a joke,” I say. “But we can approximate that value by finding the tangent line to the function Y equals the fifth root of X at X equals thirty-two.” For the next few minutes, we work on the calculus to find the equation of the tangent line, and then I go to the computer and pull up the online graphing calculator. “So let’s graph the original function, the fifth root of X, and then the equation of the tangent line we just found, Y equals one eightieth X plus eight over five.” I move the picture over to the side, where X is thirty-two, and zoom in. “Look how the graphs are overlapping. And as long as we stay close to our X value of thirty-two, the tangent line is an incredible approximation for the original function. So how could we use this to find the fifth root of thirty-three?”
A moment’s pause, and then Jim calls out, “Plug thirty-three into the equation of the tangent line.”
Yes.And this is the moment I live for. The moment when it clicks. “Exactly. And THAT is a computation we can do without a calculator. Granted, it’ll take us back to some long-division from fifth grade, but it’s still better than nothing. And even if it’s not exact, it’s really close.” I look up at the clock and notice that there are five minutes left, but there isn’t much I can really accomplish at this point. “Let’s end here for today. I’ll see you all on Wednesday.”
The students gather their belongings, and I log out of the computer. As I’m packing up my backpack with my markers and iPad, a hush comes over the room.
“What’s going on?” I ask as I lift my head, then see Claire standing in the doorway. My treacherous heart leaps at the sight of her with her auburn hair around her shoulders, wearing a green floral dress. I told her last year that green looks amazing on her, and I’m holding on to that right now.
Stop it, Ryan.No matter what I’m trying to tell myself—that I need to move on—my brain has not relayed the message to my heart.
“Hey, Professor Beaumont,” I say casually, raising a hand in greeting.
“Hey, Ryan,” she says back. She glances at the girls in the front row, who I now notice are sending daggers with their eyes.
Don’t worry, girls. I’m not the one she’s interested in.
She looks back at me. “You’re done for today, right?”
I nod. “All done teaching.”
“Can we talk? About, um…” She glances at the girls and back at me again. “About the tutoring center.”
Ugh. Not again. “Oh. Sure.”
My polite students are waiting at the doorway to leave since Claire is blocking the path. She notices and awkwardly steps back. “I’ll just, uh, wait for you. At the office.”
