Flügel Mathematics

Appearance

Application of Derivatives & Monotonicity ​

Differential Geometry Viewpoint

The derivative $f'(x_0)$ geometrically represents the instantaneous slope of the tangent line to the curve $y = f(x)$ at $(x_0, f(x_0))$. The second derivative $f''(x_0)$ governs curvature and local convexity.

1. Tangents, Normals & Length of Subtangent/Subnormal ​

Let P(x1,y1) be a point on y=f(x) with slope m=dydx|P.

  • Equation of Tangent: y−y1=m(x−x1)
  • Equation of Normal: y−y1=−1m(x−x1) (m≠0)
  • Length of Tangent: |y11+m2m|
  • Length of Normal: |y11+m2|
  • Length of Subtangent: |y1m|
  • Length of Subnormal: |y1m|

2. Rolle's & Lagrange's Mean Value Theorems ​

Rolle's Theorem

If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that: $$f'(c) = 0$$ Between any two real roots of $f(x) = 0$, there lies at least one real root of $f'(x) = 0$.

Lagrange's Mean Value Theorem (LMVT)

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c \in (a, b)$ such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$

3. Monotonicity & Extrema ​

  • Strictly Increasing on I: f′(x)>0 for all x∈I (can be 0 at isolated points).
  • Strictly Decreasing on I: f′(x)<0 for all x∈I.

First Derivative Test for Local Extrema ​

  • Local Maximum at x0: f′(x) changes sign from positive to negative as x increases through x0.
  • Local Minimum at x0: f′(x) changes sign from negative to positive as x increases through x0.

Second Derivative Test ​

  • If f′(x0)=0 and f″(x0)<0⟹ Local Maximum at x0.
  • If f′(x0)=0 and f″(x0)>0⟹ Local Minimum at x0.

4. Concavity & Point of Inflection ​

  • f″(x)>0⟹ Concave Upwards (Convex ⌣, tangent lies below curve).
  • f″(x)<0⟹ Concave Downwards (Concave ⌢, tangent lies above curve).
  • Point of Inflection: A point where f″(x) changes sign and tangent exists.