Calculus I

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Mind Map on Calculus I, created by GraceEChem on 08/12/2014.
GraceEChem
Mind Map by GraceEChem, updated more than 1 year ago
GraceEChem
Created by GraceEChem about 9 years ago
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Resource summary

Calculus I
  1. Limits
    1. Infinite limits
      1. Divide top and bottom by biggest power
        1. 1/x^n = 0
        2. Finite limits
          1. Continuous, plug in constant
            1. 0/0 is your answer?
              1. Factor and Simplify
        3. Derivatives
          1. Product Rule
            1. (fg)' = f'g+g'f
            2. Quotient Rule
              1. (f/g)' = (f'g - g'f) / g^2
              2. Power Rule for Functions
                1. (f^n)' = nf^(n-1)f'
                2. Chain Rule
                  1. dy/dx = (dy/du)(du/dx)
                    1. for y=y(u) and u=u(x)
                    2. [f(g(x)]' = f'(g(x))g'(x)
                    3. Finding 2nd Derivative
                      1. just take the derivative of the derivative
                      2. f(x) = e^x, f'(x) = e^x
                        1. f(x) = ln x, f'(x) =1/x
                          1. g(x) = loga(x), g'(x) = (1/ln a)(1/x)
                            1. f'(x) = ln(a)a^x
                            2. Chain Rule and Logs
                              1. g(x) = ln(f(x))
                                1. f'(x)/f(x)
                              2. ln(x) + 1/x (x')
                            3. Marginal Analysis/Linear Approximation
                              1. f(x+delta x) ~= f(x) + delta x(f'(x))
                                1. or Marginal Cost = C'(x)
                                2. Implicit Differentiation
                                  1. Treat x as x and y as f(x), give derivative of y w/respect to x
                                    1. Differentiate both sides w/respect to x with y as f(x)
                                      1. Chain rule, differentiating terms with y
                                        1. Solve for dy/dx in terms of x and y
                                    2. Increasing/Decreasing Functions
                                      1. Use 1st Derivative
                                        1. Test points @ f'(x) = 0
                                          1. Plug test points into f'(x)
                                            1. And where f'(x) does not exist
                                              1. CRITICAL POINTS!
                                            2. Concavity
                                              1. Use f''(x)
                                                1. Test points @ f''(x) = 0
                                                  1. And where f''(x) does not exist
                                                    1. Inflection point where concavity changes
                                              2. Positive/Negative
                                                1. Test points on f(x)
                                              3. Optimization
                                                1. x = c is critical point if f'(c) = 0
                                                  1. x = c relative min/max if f'(c) changes sign
                                                    1. x = c abolute min/max
                                                    2. Closed Interval
                                                      1. find where f'(x)=0
                                                        1. plug those points into f(x)
                                                          1. compare against interval points for absolute max/min
                                                      2. Open Interval
                                                        1. find one critical point, where f'(x)=0
                                                          1. this shows concavity
                                                      3. Exponents
                                                        1. Compound interest
                                                          1. Yearly
                                                            1. P(t) = P(1+r)^t
                                                            2. Monthly
                                                              1. P(t) = P(1+r/12)^12t
                                                              2. Continuously
                                                                1. P(t) = Pe^rt
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