A Line Stands In For The Curve, And The Bend Says Which Side It Lands On

You'll learnto build a linearization with all three of its pieces in the right places, and to decide whether the answer came out too high or too low from the sign of the second derivative rather than the first.

Magnify a differentiable curve enough and it stops looking curved. That is the whole content of the derivative, and it means a tangent line can stand in for the function near the point of tangency. Building the line takes three pieces — the height at the center, the slope at the center, and the step you took from the center — and the third is the one that goes missing. Then there is the part worth the most and skipped the most: whether the answer came out too high or too low. That is settled by the bend, never by whether the function is rising.

8 STEPS · 6 QUICK CHECKS · THE STEP IS x − a, NOT x · THE CENTER MUST BE NEAR AND EXACT · f″ DECIDES THE SIDE, NOT f′ · v1

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, f(x) = 6 + 1.5(x − 2) + 0.5(x − 2)³, for x from 0 to 4. It is increasing everywhere. It bends downward on the left half and upward on the right half, and both of the centers we linearize from have exactly the same slope, 3.

The question

A tangent line can stand in for the curve near the point of tangency. What are the pieces of that line, and is the number it produces too big or too small?

Watch for

The zoom, where the plot window never changes and the curve becomes the line anyway. And the two tangents at the end: same slope, same direction of travel, opposite verdicts.

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