suppose we want a linear approximation for sin(0.3). Look at the function sinx, where x = 0.3 We want to pick a value for a to approximate x, a value for which we know the sin. We know sin0 = 0, so choose a = 0, which is close to x =0.3.
Now use the formula L(x) (linear approximation as function of x) = f(a) + f'(a)(x-a)
f(0) = 0
f'(x) = cosx, so f'(a) = f'(0) = 1
L(x) = 0 + 1(0.3 - 0)
= 0 + 0.3
The value of sin(0.3) = .295. so the approximation is close. Note that the value of x = 0.3 is in radians