Linearisation and Differentials

Using the tangent line as the function.

12 problems · about 21 minutes · CHA-3.F

NameDate

Figure — use these to answer the problems

See how tightly the tangent hugs the curve near the point, and how fast it drifts away.

y = 1√x + 0

a, coefficient1

Point of tangency4

Figure 1

Straightforward practice. Get the method working first.

5 problems

L(x) = 4 + 0.5(x − 16). What is L(16.4)?
Answer
2
What determines whether an approximation is an over- or underestimate?
- The concavity, given by the sign of f double prime.
- The sign of the slope.
L(x) = 3 + 2(x − 1). What is L(1.5)?
Answer
f(9) = 3 and f′(9) = 1/6. Estimate f(9.6).
Answer
A curve is concave up. Over- or underestimate? 1 over, 2 under.
Answer

The same ideas with more to keep track of.

3 problems

L(x) = 2 + 0.25(x − 4). What is L(4)?
Answer
dy = f′(x) dx with f′(x) = 6 and dx = 0.5. What is dy?
Answer
Which sign of f double prime gives an overestimate? 1 positive, 2 negative.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Which choices improve a linear approximation?
- A base point close to the target
- A curve that is nearly straight there
- Moving far from the base point
- A sharply bending curve
- Tick every box that applies.
L(x) = 5 + 3(x − 2). What is L(2.1)?
Answer
The Estimate
L(x) = 2 + 0.25(x − 4). What is L(4.2)?
Answer
The Direction
A curve is concave down. Is the tangent estimate an over- or underestimate? 1 over, 2 under.
Answer