Calculus

Calculus

1. Limits

   • lim(x→a) f(x) may not be equal to f(a)
   • value that a function approaches as the input approaches some value
   • exists only if both the LHS & RHS limits are defined
   • properties
     • lim(x→a) [k f(x)] = k lim(x→a) f(x)
     • lim(x→a) [f(x)±g(x)] = lim(x→a) f(x) ± lim(x→a) g(x)
     • lim(x→a) [f(x)g(x)] = lim(x→a) f(x) ⨉ lim(x→a) g(x)
     • lim(x→a) [f(x) ÷ g(x)] = lim(x→a) f(x) ÷ lim(x→a) g(x)
   • limits at infinity gives horizontal/ slant asymptotes

2. Differentiation

   • rate of change of single-variable functions (within IBDP)
   • methods of differentiation
     • rules of differentiation
       • product rule
       • addition rule
       • power rule
       • subtraction rule
       • quotient rule
       • chain rule
     • first principles (formula in formula booklet)
     • implicit differentiation
   • function is differentiable if dirivative can be found
   • second derivative = diffrentiate twice
   • nth derivative = differentiate n times
   • graphical interpretation
     • gradient of the tangent to the curve
       • normal is perpendicular to tangent
     • first derivative: increasing when f'(x)>0, decreasing when f'(x)<0
       • first derivative test
         • check for changes of sign when f'(x)=0
           • from + to - maxima
           • from - to + minima
           • no change in sign: horizontal point of inflection
     • second derivative: concave up when f''(x)>0, concave down when f''(x)<0
       • second derivative test
         • After finding f'(x)=0, f''(x)>0 indicates minima, f''(x)<0 indicates maxima

3. Integration

   • antiderivative
   • types of integration
     • indefinite integral
       • answer requires "+c", where c is the constant of integration
     • definite integral
       • evaluate at upper and lower limits
       • gives (net) area under the curve
       • volume of revolution = π∫[f(x)]² dx
   • methods of integration
     • formula in formula booklet
     • by inspection
     • double angle identities
       • to reduce power of sin/cos
       • to simplify integrand
     • by parts
     • by partial fractions
       • where the denominator of the integrand can be factorised
     • substitution
       • u substitution
       • trigonometric substitution
   • properties of integration
     • ∫kf(x) dx = k∫f(x) dx
     • ∫[f(x)±g(x)]dx = ∫f(x)dx ± ∫g(x)dx

4. Differential equations

   • first-order linear
     • multiply both sides by integrating factor
   • homogeneous
     • substitute v=y/x
   • separable
   • other first order differential equations
     • Euler's method

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