Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Local linearity: zoom in on a differentiable function and its graph looks like its tangent line.
- Linearization at a: L(x) = f(a) + f′(a)(x − a). Use f(x) ≈ L(x) for x close to a.
- Change form: change in f ≈ f′(a) × (change in x).
- Choose a where f(a) and f′(a) are easy to find, as close as possible to the x you want.
- The error grows as x moves away from a.
- No linearization at a point where f is not differentiable (a corner or a cusp).
Key relationships
| Near x = a | Tangent line is | Estimate is |
|---|---|---|
| f″ > 0 (graph bends upward) | below the curve | an underestimate |
| f″ < 0 (graph bends downward) | above the curve | an overestimate |
| f″ changes sign at a | on different sides on the left and right | check each side separately |
Common choices: ∛x near 8 or 27, √x near a perfect square, 1/x near an easy whole number.
Assumptions behind the method
- f is differentiable at a, so the tangent line exists.
- x is close to a; “close” depends on how quickly f′ changes.
- To judge over or under, you need the sign of f″ (or the bend of the graph) on the interval between a and x.
Mistakes to avoid
- Using f′(x) or f′ at the wrong point. The slope is f′(a).
- Dropping f(a), so that only the change is reported.
- Using x in place of (x − a).
- Swapping over and under. Bending upward means the line is below: underestimate.
- Using the sign of f′ to decide over or under.
- Writing “=” instead of “≈”, or forgetting units in context.
Quick self-check
- f(1) = 4 and f′(1) = −3. Estimate f(1.1). (L(1.1) = 4 − 3(0.1) = 3.7)
- f″(x) > 0 near x = a. Is L(x) too high or too low? (Too low: an underestimate.)
- Estimate √49.7 using a = 49. Over or under? (7 + (1/14)(0.7) = 7.05; √x bends downward, so 7.05 is an overestimate. Actual ≈ 7.0498.)
Next: practice questions.