Applications of Differential Calculus

Last updated Jul 29, 2023 Edit Source

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• The derivative of a function measures the gradient of the tangent at any point of the original function

  • We can use differentiation to find the equation of the tangent line at any point of a function
    • The equation of a straight line is $y = mx + c$
      • For the tangent, $m$ equals the derivative

• The tangent equation at point $(x_1,\\ y_1)$: → $y - y_1 = f’(x_1)(x - x_1)$

  • e.g. Let $x_1 = 1$, $f(x) = x^2 + 3x$
    • $\therefore \\ y_1 = 4$, $f’(x) = 2x + 3$
    • The tangent at $x = 1$ → $y - 4 = (2(1) + 3)(x - 1)$ → $y = 5x - 1$

• The normal is the line which passes through a point and is perpendicular to the tangent at that point

  • Thus the gradient of the normal is the negative reciprocal of the gradient of the tangent
    • If the tangent has gradient $m$ → the normal has gradient $-\frac{1}{m}$
  • We can calculate the equation of the normal by using the same equation for the tangent while changing the gradient

• Points where the derivative = 0 are called stationary points
  • A point $(a, f(a))$ is considered to be a stationary point if $f’(a) = 0$
    • The tangent to these points are parallel to the $x$-axis
  • Points $A$, $B$ and $C$ are stationary points below: StationaryPoints
  • Turning points are stationary points
    • Sine and cosine functions and straight lines ($y = a$) have infinite stationary points

• Cambridge - 18D
• There are 3 types of stationary points: TypesOfStationaryPoints
  • Point $A$ is known as a local maximum point
    • $f’(x) > 0$ to the left of $A$
    • $f’(x) < 0$ to the right of $A$
  • Point $B$ is known as a local minimum point
    • $f’(x) < 0$ to the left of $A$
    • $f’(x) > 0$ to the right of $A$
  • Point $C$ is known as a stationary point of inflection
    • Either:
      • $f’(x) < 0$ to the left and right of $C$
      • $f’(x) > 0$ to the left and right of $C$
  • Points $A$ and $B$ are known as turning points
• For a function defined on an interval:
  • The actual maximum value of the function is called the absolute maximum
  • The actual minimum value of the function is called the absolute minimum

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