Barrons AP Calculus

(a)

(b)

(c)

(d)

At  a   =   0,  sin x   sin (0) +   cos (0)(x   −   0)      0    +  1    · x x

At  a   =   1,  2x^3    −   3x  −1  +   3(x −   1)      3 x −   4

Example 31 __

Using the tangent lines obtained in Example 30 and a calculator, we evaluate
each function, then its linear approximation, at the indicated x-values:

Example 31 shows how small the errors can be when tangent lines are used for
approximations and x is near a.

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 .