Cogito
AP Calculus AB · Chapter 4 · Lesson 2
Linearisation and Differentials
Using the tangent line as the function.
12 problems · about 21 minutes · CHA-3.F
What this lesson teaches
The student approximates function values with a tangent line and identifies over- and underestimates.
- L(x) = f(a) + f′(a)(x − a) uses the tangent as a stand-in.
- Concave down overestimates; concave up underestimates.
- Accuracy falls away as you move from the base point.
Warm Up
Straightforward practice. Get the method working first.
5 problemsL(x) = 4 + 0.5(x − 16). What is L(16.4)?
Answer 4.2
Why 4.2.
What determines whether an approximation is an over- or underestimate?
Answer The concavity, given by the sign of f double prime.
Why Concavity decides it.
L(x) = 3 + 2(x − 1). What is L(1.5)?
Answer 4
Why 3 + 1.
f(9) = 3 and f′(9) = 1/6. Estimate f(9.6).
Answer 3.1
Why 3 + 0.6 ÷ 6.
A curve is concave up. Over- or underestimate? 1 over, 2 under.
Answer 2
Why The curve sits above its tangent.
Build It Up
The same ideas with more to keep track of.
3 problemsL(x) = 2 + 0.25(x − 4). What is L(4)?
Answer 2
Why They meet there.
dy = f′(x) dx with f′(x) = 6 and dx = 0.5. What is dy?
Answer 3
Why 6 × 0.5.
Which sign of f double prime gives an overestimate? 1 positive, 2 negative.
Answer 2
Why Concave down.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich choices improve a linear approximation?
Answer A base point close to the target; A curve that is nearly straight there
Why Bending is what defeats a line.
L(x) = 5 + 3(x − 2). What is L(2.1)?
Answer 5.3
Why 5 + 0.3.
The Estimate: L(x) = 2 + 0.25(x − 4). What is L(4.2)?
Answer 2.05
Why 2.05.
The Direction: A curve is concave down. Is the tangent estimate an over- or underestimate? 1 over, 2 under.
Answer 1
Why An overestimate.