L2 Calculus

L2 Calculus

1. Differentiation

   • Finding the gradient or rate of change at a point (Differentiate, substitute in the x value)
     • Find the equation of a tangent (Use y-y1=m(x-x1)
       • Find the equation of a normal (Use m1xm2=-1 and y-y1=m(x-x1)
     • Find velocity given distance
     • Find acceleration given velocity or distance
   • Find the point with a certain gradient (Differentiate, set equal to gradient given, solve for x, substitute to find y / f(x)
     • Find where increasing / decreasing (Differentiate, solve f'(x)>0 or f'(x) <0)
     • Finding turning points (Differentiate, solve f'(x)=0)
       • Finding maximums and minimums (Find equation to maximise or minimise, differentiate, solve f'(x)=0 to find where max/min occurs, substitute into f(x) to find max or min
         • Prove a point is max or min (Find f''(x), if f''(x)>0 then a min, if f''(x)<0 then a max, if f''(x)=0 we cannot tell
         • Find max or min distance (where v=0)
         • Find max or min velocity (where a=0)

2. Anti-differentiation

   • Finding the function from the gradient function and a point (Anti-differentiate, substitute in point to find c)
     • Find a point on the orginal cuve or function (Find original function, substitute or solve to find point
     • Find velocity given acceleration
     • Find distance given velocity or acceleration

3. Sketching functions

   • Sketch function from gradient function (x-intercepts become turning points, above y-axis is increasing, below y-axis is decreasing
     • Sketch velocity function from acceleration function
     • Sketch distance function from velocity function
   • Sketch gradient function from function (Turning points becomes x-intercepts, Increasing is above y-axis, decreasing is below y-axis
     • Sketch acceleration function from velocity function
     • Sketch velocity function from distance function

4. Kinematics