Cogito
Calculus · Chapter 5 · Lesson 3
Linear Approximation
Use the tangent line instead of the curve.
12 problems · about 21 minutes · CHA-3.F
What this lesson teaches
The student uses a tangent line to approximate function values and explains when the estimate is reliable.
- L(x) = f(a) + f′(a)(x − a) uses the tangent line 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.
Why does the estimate worsen far from the base point?
Answer The curve bends away from its tangent line.
Why The curve bends and the line does not.
L(x) = 3 + 2(x − 1). What is L(1.5)?
Answer 4
Why 3 + 2 × 0.5.
f(9) = 3 and f′(9) = 1/6. Estimate f(9.6).
Answer 3.1
Why 3 + 0.6 ÷ 6.
f(x) = √x. What is f(25)?
Answer 5
Why The exact root.
Build It Up
The same ideas with more to keep track of.
3 problemsA curve is concave up. Is the estimate high or low? 1 high, 2 low.
Answer 2
Why The curve sits above its tangent.
L(x) = 2 + 0.25(x − 4). What is L(4)?
Answer 2
Why The line meets the curve there.
dy = f′(x) dx. With f′(x) = 6 and dx = 0.5, what is dy?
Answer 3
Why 6 × 0.5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich choices make a linear approximation more accurate?
Answer Choosing a base point close to the target; A curve that is nearly straight near the point
Why Approximation is a local tool, and bending is what defeats it.
L(x) = 5 + 3(x − 2). What is L(2.1)?
Answer 5.3
Why 5 + 3 × 0.1.
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 too high or too low? 1 high, 2 low.
Answer 1
Why Too high.