Florida B.E.S.T. MA.912.C.3.2

MathGrades 9–12Apply derivatives to solve problems.

The Standard

Find an equation for the tangent line to a curve at a point and use it to make local linear approximation.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students compute the derivative at a chosen input and use it as the slope through the corresponding point on the curve. They use the resulting line to estimate nearby function values.
What Mastery Looks Like: Given a function and input a, a student finds f(a) and f'(a), then writes y − f(a) = f'(a)(x − a). The student uses the line to estimate a nearby value and explains why estimates usually weaken farther from a.
Common Misconceptions: Students may use f(a) instead of f'(a) as the slope, or substitute the wrong input into the derivative. They may write a secant line, make sign errors in point-slope form, or assume the estimate stays accurate far from the contact point.
How to Assess It: Give this exit ticket: For f(x) = √x at x = 4, write the tangent line and use it to estimate √4.1.

Lesson moves

Plot a curve on graph paper, mark a point, use a ruler to draw the tangent, then compare its slope with the derivative.
Ask: Why does the tangent estimate work near the contact point but become less reliable farther away?
Match cards showing functions, points, derivative values, tangent equations, and nearby estimates, then justify each completed set.
Use a tank’s current water level and rate of change to predict its level two minutes later.

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