Common Core · AP Calculus AB
CHA-3.F
AP Calculus AB, Unit 4 — Approximate a value of a function using local linearity and linearization.
A note on this mapping: The Common Core has no calculus standards; this is the AP Calculus AB learning objective.
The 2 skills that cover it
- Linear Approximation & DifferentialsReplace a curve by its tangent line where you stand: write the linearization, use it to estimate a root or a power, work with differentials and propagated error, and say whether the estimate is high or low.Read the lesson →
- Newton's MethodApproximate a root by sliding down tangent lines: take one step from a starting guess, write the iteration formula, run it to a given precision, and recognise the starts where it gets stuck.Read the lesson →
What a student should be able to do
- Write the linearization L(x) = f(a) + f'(a)(x - a).
- Approximate a value such as sqrt(26) or (1.02)5 with a tangent line, choosing the anchor that makes it accurate, or from a reading and a rate given in a table, a graph or a context.
- Use the differential dy to estimate a change and a propagated or percentage error.
- Use concavity to decide whether a linear approximation is an over- or under-estimate.
- Take one Newton step x1 = x0 - f(x0)/f'(x0) exactly.
- Write the iteration formula for a given function.
- Iterate to a stated precision, read a table of successive estimates, and judge how many decimal places an estimate has earned.
- Choose a function whose root is the number wanted, and recognise a start that gets stuck.