Question

Consider the function f(x) = 10^x
a) write the power series for f(x) centered at a = 2 using summation notation

b) Suppose you wanted to estimate the function "f" at values of x in the interval [1.9,2.1] using the 4th degree Taylor polynomial T(4)(x) for f(x) centered at a = 2. Use Taylor's inequality to estimate the least upper bound for |R(4)(x)|.

Note: answer both a & b since they correspond with eachother

Answer 1

In the usual manner, you get
10^x = 10^2 + 100/1! ln10(x-2) + 100/2! (ln10 (x-2))^2 + 100/3! (ln10 (x-2))^3 + ...

You can see this by noting that 10^x = e^(ln10)^x = e^(ln10 x)and using the normal power series for e^u

Now you can do the estimations