Question

Exercise 1:

Let f(x) = sin(2x).
(a) Find L(x) at a = 0.
(b) Use L(x) to approximate sin(0.2).
(c) Estimate the error in the above approximation.
Exercise 2:
Let p > 0.

Answer 1

To find L(x) at a = 0, L(x) = sin(0) = 0. To approximate sin(0.2) using L(x), L(0.1) = sin(0.2) = 0.1987. The estimated error in the approximation is 0.0013.

Step-by-step explanation:

(a) To find L(x) at a = 0, we substitute a = 0 into the function f(x) = sin(2x). L(x) = sin(2(0)) = sin(0) = 0.

(b) To approximate sin(0.2) using L(x), we substitute x = 0.1 into L(x). L(0.1) = sin(2(0.1)) = sin(0.2) = 0.1987.

(c) To estimate the error in the approximation, we calculate the absolute difference between the actual value of sin(0.2) and the approximation using L(x). Error = |0.1987 - sin(0.2)| = 0.0013.