Barrons AP Calculus - David Bock

EXAMPLE 32

A very useful and important local linearization enables us to approximate (1 + x) k by 1 + kx for

k any real number and for x near 0. Equation (1) yields

Then, near 0, for example,

EXAMPLE 33

Estimate the value of at x = 0.05.

SOLUTION: Use the line tangent to at x = 0; f (0) = 3.

so f ′(0) = 6 ; hence, the line is y = 6x + 3.

Our tangent-line approximation, then, is

At x = 0.05, we have f (0.05) ≈ 6(0.05) + 3 = 3.3.

The true value, to three decimal places, of when x = 0.05 is 3.324; the tangent-line

approximation yields 3.3. This tells us that the curve is concave up, lying above the tangent line

to the curve near x = 0. Graph the curve and the tangent line on [−1, 1] × [−1, 6] to verify these

statements.

Approximating the Change in a Function.
Equation (1) above for a local linear approximation also tells us by about how much f changes when
we move along the curve from a to x: it is the quantity f ′(a)(x − a). (See Figure N4–20.)

EXAMPLE 34

By approximately how much does the area of a circle change when the radius increases from 3 to

3.01 inches?

SOLUTION: We use the formula A = πr^2. Then Equation (1) tells us that the local linear

approximation for A(r), when A is near 3, is

A(3) + A ′(3)(r − 3).

Here we want only the change in area; that is,

A ′(3)(r − 3) when r = 3.01.

Since A ′(r) = 2πr, therefore A ′(3) = 6π; also, (r − 3) = 0.01, so the approximate change is (6π)

(0.01) 0.1885 in.^2 The true increase in area, to four decimal places, is 0.1888 in.^2