A Line Stands In for the Curve, and the Bend Picks the Side

You'll learnto build a tangent-line approximation from its three pieces, to pick a center that is near and exactly known, and to decide whether the answer is too high or too low from the bend of the curve, not its slope.

Zoom in far enough on a smooth curve and it looks like a straight line, so a tangent line can stand in for the curve near the point where it touches. Ten short steps; press Next for each one.

10 STEPS · 6 QUICK CHECKS · THE STEP IS x − a, NOT x · THE CENTER MUST BE NEAR AND EXACT · f″ DECIDES THE SIDE, NOT f′ · v2

f(x) = 6 + 1.5(x − 2) + 0.5(x − 2)³ ⇒ L(x) = f(a) + f′(a)(x − a)

Before you start

What you're looking at

One curve, rising the whole way, bending down on the left and up on the right. A straight line will be laid against it at one point.

The question

What are the pieces of that line, and is the number it gives too big or too small?

Step 1 / 10