Calculus I

Calculus I

1. Limits

   • Infinite limits
     • Divide top and bottom by biggest power
     • 1/x^n = 0
   • Finite limits
     • Continuous, plug in constant
       • 0/0 is your answer?
         • Factor and Simplify

2. Derivatives

   • Product Rule
     • (fg)' = f'g+g'f
   • Quotient Rule
     • (f/g)' = (f'g - g'f) / g^2
   • Power Rule for Functions
     • (f^n)' = nf^(n-1)f'
   • Chain Rule
     • dy/dx = (dy/du)(du/dx)
       • for y=y(u) and u=u(x)
     • [f(g(x)]' = f'(g(x))g'(x)
   • Finding 2nd Derivative
     • just take the derivative of the derivative
   • f(x) = e^x, f'(x) = e^x
     • f(x) = ln x, f'(x) =1/x
     • g(x) = loga(x), g'(x) = (1/ln a)(1/x)
       • f'(x) = ln(a)a^x
     • Chain Rule and Logs
       • g(x) = ln(f(x))
         • f'(x)/f(x)
     • ln(x) + 1/x (x')

3. Marginal Analysis/Linear Approximation

   • f(x+delta x) ~= f(x) + delta x(f'(x))
   • or Marginal Cost = C'(x)

4. Implicit Differentiation

   • Treat x as x and y as f(x), give derivative of y w/respect to x
   • Differentiate both sides w/respect to x with y as f(x)
     • Chain rule, differentiating terms with y
       • Solve for dy/dx in terms of x and y

5. Increasing/Decreasing Functions

   • Use 1st Derivative
     • Test points @ f'(x) = 0
       • Plug test points into f'(x)
       • And where f'(x) does not exist
       • CRITICAL POINTS!
   • Concavity
     • Use f''(x)
       • Test points @ f''(x) = 0
         • And where f''(x) does not exist
           • Inflection point where concavity changes
   • Positive/Negative
     • Test points on f(x)

6. Optimization

   • x = c is critical point if f'(c) = 0
     • x = c relative min/max if f'(c) changes sign
     • x = c abolute min/max
   • Closed Interval
     • find where f'(x)=0
       • plug those points into f(x)
         • compare against interval points for absolute max/min
   • Open Interval
     • find one critical point, where f'(x)=0
       • this shows concavity

7. Exponents

   • Compound interest
     • Yearly
       • P(t) = P(1+r)^t
     • Monthly
       • P(t) = P(1+r/12)^12t
     • Continuously
       • P(t) = Pe^rt