Approximations at 0 for Sine, Cosine and Exponential Functions

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2019
Views:

498

Pages:

1
Author:

espansaxkj

Approximations at 0 for Sine, Cosine and Exponential Functions Here is a list of several linear approximations which you may want to memorize. Half the work of memorizing a linear approximation is memorizing the derivative of a function at a base point, so memorizing these formulas should improve your knowledge of derivatives. To make things as simple as possible, we always use base point x0 = 0 and assume that x ≈ 0. Then our general formula becomes: f (x) ≈ f (0) + f � (0)x. Remember that when x is not near zero, this approximation probably won’t work. (Later we’ll discuss exactly how close x has to be to zero; this is partly a matter of intuition and is very important in applications.) We want to ﬁnd linear approximations for the functions sin x, cos x and ex when x is near 0. We’ll start by building a table of values of f � (x), f (0), and f � (0); from these we can “read oﬀ” the linear approximations. f (x) f � (x) f (0) f � (0) sin x cos x 0 1 cos x − sin x 1 0 ex ex 1 1 We can now plug the values for f (0) and f � (0) into our formula f (x) ≈ f (0) + f � (0)x to get linear approximations for these functions: 1. sin x ≈ x (if x ≈ 0) (see part (a) of Fig. 1) 2. cos x ≈ 1 (if x ≈ 0) (see part (b) of Fig. 1) 3. ex ≈ 1 + x (if x ≈ 0) y=x y=1 sin(x) cos(x) (a) (b) Figure 1: Linear approximations to sine and cosine at x = 0. 1