# Calculus I

### Description

Mind Map on Calculus I, created by GraceEChem on 08/12/2014.
Mind Map by GraceEChem, updated more than 1 year ago
 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. 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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