MATH137 Lecture Notes - Lecture 8: Maxima And Minima, Mean Value Theorem, Intermediate Value Theorem

• School: University of Waterloo
• Department: Mathematics
• Course: MATH137
• Professor: Catherine Briggs

Document Summary

Math 137 week 8 notes dr. paula smith y. 3. 10 the equation of a line tangent to a curve f(x) at point a is y = f(a) + f (a)(x a). When x is near a, y f(x). We can rewrite the approximation as f(x + x) f(x) + f (x)(x + x x) = f(x) + f (x) x, and recognize that y = f(x + x) f(x) f (x) x. In the limit, we obtain dy = f (x) dx. Both dx and dy are called a differential, and their meaning is an infinitesimal change in the x (or y) direction. The meaning of x and y is a small but determinable change in the x (or y) direction. Often we state dx = x; then dy is the amount that the tangent line rises or falls and y is the amount that the curve rises or falls when x is changed by a small amount dx.