微积分的一些相关定理

高数学了都忘了，复习下

维基百科上称此为微积分第一基本定理（First Fundamental Theorem of Calculus）

Mean value theorem中值定理

Let f : [a, b] → R be a continuous function on the closed interval [a,b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

cite from

http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/grad/grad.html

Directional Derivatives

For a function z=f(x,y), the partial derivative with respect to x gives the rate of change of f in the x direction and the partial derivative with respect to y gives the rate of change of f in the y direction. How do we compute the rate of change of f in an arbitrary direction?

The rate of change of a function of several variables in the direction u is called the directional derivative in the direction u. Here u is assumed to be a unit vector. Assuming w=f(x,y,z) and u=<u_1,u_2,u_3>, we have

Hence, the directional derivative is the dot product of the gradient and the vector u. Note that if u is a unit vector in the x direction, u=<1,0,0>, then the directional derivative is simply the partial derivative with respect to x. For a general direction, the directional derivative is a combination of the all three partial derivatives.