Differentiation overview year 12

Differentiation overview year 12

1. Finding the gradient function at any point x (gradient of the tangent)

   • Finding stationary points
     • Nature
       • Max f"(x)<0
       • Min f"(x) > 0
       • Inflexion f"(x) = 0
   • Finding equations of tangents and normals
     • Require point (coordinate) and gradient
       • tangent gradient = f'(x) Normal gradient = -1/f'(x)
         • y-y1 = m(x - x1)
   • sketch the gradient function
     • turning points (where gradient changes from positive to negative or negative to positive)
       • x intercept (y = 0) y intercept (x = 0)
         • asymptotes (values of x for which would lead to dividing by zero)

2. Basic e.g. dy/dx

3. Modelling

   • Optimisation problems -use volume and surface area of common solids e.g. prisms, spheres
     • usually substitute one formula into another
       • Find maximum /minimum by first differentiating and finding x values for which the stationary point occurs
         • Substitute x value into original formula for quantity of interest to find its min/max value
           • Justify the max/min using second derivative

4. By first principles

   • Consider the gradient of the function as delta x approaches zero