[advanced mathematics] [3] Application of differential mean value theorem and derivative

1. Differential mean value theorem

1.1 Rolle's theorem

1.1.1 Fermat lemma

1.1.2 Rolle's theorem

1.2 Lagrange mean value theorem ( Differential mean value theorem )

1.3 Cauchy mean value theorem

2. The law of lophida

2.1 Lobida theorem 1【0/0】

Infinitive

2.2 Lobida theorem 2【∞/∞】

2.3 Type close 0/0 or ∞/∞

2.* matters needing attention

3. Taylor formula

Taylor formula purpose :
In order to improve the accuracy of approximate expressions , Use higher degree polynomials to approximate functions .

3.1 Taylor Mean Value Theorem 1

3.2 Taylor Mean Value Theorem 2

3.3 McLaughlin's formula

4. Monotonicity of function and concavity and convexity of curve

4.1 Monotonicity of function

4.2 Concavity and convexity of curve and inflection point

5. Extremum, maximum and minimum of function

5.1 Max min definition

5.2 Theorem

The derivative of the function is 0 The point of a function is called the stagnation point of a function

5.3 Steps for finding extreme points

6. The depiction of function graph

7. Curvature

7.1 Arc differential formula

7.2 Curvature

The curvature at each point on the circle is equal to the radius a Reciprocal 1/a

The curvature of the parabola at the vertex is the largest

7.3 Curvature circle and radius of curvature

8. Approximate solution of the equation