MATH 104 Lecture Notes - Quadratic Function

• School: UBC
• Department: Mathematics
• Course: MATH 104
• Professor: Jacob Levy

Document Summary

These notes cover the material of the classes dedicated to approxi- mation. Using derivatives, we can nd good approximations for how functions behave near a point where it is easy to nd their value. In cases like this, we can apply the derivative formula in reverse. Since f(cid:48)(a) = lim x a f (x) f (a) x a it follows that when x is close to a, we have f(cid:48)(a) f (x) f (a) x a. Now we get rid of the fraction and obtain f (x) f (a) f(cid:48)(a)(x a) and then moving f (a) to the other side, we have f (x) f (a) + f(cid:48)(a)(x a) Let f (x) be a di erentiable function, and suppose that x a is small. Given f (a) and f(cid:48)(a), we call f (a) + f(cid:48)(a)(x a) the linear approxi- mation of f (x). We will postpone the question of how good the approximation after some examples.