Geometrical applications of the derivative (angle between curves, tangent and normal) - Increasing and decreasing functions - Maxima and Minima (single variable functions) using second order derivative only physical application - Rate Measure

 Geometrical applications of the derivative:

  • The derivative can be used to find the slope of a tangent line to a curve at a given point.
  • The derivative can also be used to find the equation of a normal line to a curve at a given point, which is a line perpendicular to the tangent line at that point.
  • The angle between two curves at a point of intersection can be found using the slopes of the tangent lines to each curve at that point.

Increasing and decreasing functions, maxima and minima:

  • A function is increasing if its derivative is positive and decreasing if its derivative is negative.
  • A function has a local maximum at a point where the derivative changes sign from positive to negative, and a local minimum at a point where the derivative changes sign from negative to positive.
  • A function has a global maximum or minimum at the highest or lowest point, respectively, on the entire domain of the function.
  • The second order derivative can be used to determine whether a local maximum or minimum is a maximum or minimum, respectively. Specifically, a local maximum is a maximum if the second order derivative is negative at that point, and a local minimum is a minimum if the second order derivative is positive.


Physical applications of the derivative:

  • The derivative can be used to measure the rate of change of a physical quantity, such as velocity, acceleration, or temperature.
  • The derivative of displacement with respect to time gives velocity, the derivative of velocity with respect to time gives acceleration, and so on.
  • The integral of the derivative can be used to find the total change in a physical quantity over a given interval, such as the total distance traveled by an object or the total heat energy absorbed by a material.

Important Formulas to Remember in these topic

Derivative Formulas

Geometrical Applications of the Derivative

  • Angle between Two Curves: $$\tan\theta = \frac{dy_1/dx - dy_2/dx}{1 + dy_1/dx\cdot dy_2/dx}$$
  • Tangent and Normal to a Curve: $$y - y_0 = \frac{dy}{dx}(x - x_0) \quad \text{(Tangent)}$$ $$y - y_0 = -\frac{dx}{dy}(x - x_0) \quad \text{(Normal)}$$