4.1 Further approximations (#differentiation)

As you saw in Lesson 8, we can extend the notion of a linear approximation to a

quadratic approximation. Sometimes this improves the accuracy of the approxima-

tion.

(a) Graph both e and its linear approximation on the same plot, displaying a values

in the interval [0, 2]. What is the exact value of e0.1 and its approximate value

from your linear approximation? Compute the relative error.

(b) Find a quadratic approximation to f(x) = e near x = 0. Graph both e

and its quadratic approximation on the same plot (using the same scale as be-

fore). What is the approximate value of e0.1 from your quadratic approximation?

Compute the relative error.

(c) We can continue to add terms to our approximation. Find a formula for the

"cubic approximation" C(r) of a function g(x) around x = c. You should leave

your answer in terms of g and its derivatives.

(d) Repeat the previous two steps but for a cubic approximation using the formula

you found in the previous step. What is the approximate value of 0.1 using

your cubic approximation?