Calculus I

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Mind Map on Calculus I, created by GraceEChem on 12/08/2014.

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GraceEChem
Created by GraceEChem over 4 years ago
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Calculus I
1 Limits
1.1 Infinite limits
1.1.1 Divide top and bottom by biggest power
1.1.2 1/x^n = 0
1.2 Finite limits
1.2.1 Continuous, plug in constant
1.2.1.1 0/0 is your answer?
1.2.1.1.1 Factor and Simplify
2 Derivatives
2.1 Product Rule
2.1.1 (fg)' = f'g+g'f
2.2 Quotient Rule
2.2.1 (f/g)' = (f'g - g'f) / g^2
2.3 Power Rule for Functions
2.3.1 (f^n)' = nf^(n-1)f'
2.4 Chain Rule
2.4.1 dy/dx = (dy/du)(du/dx)
2.4.1.1 for y=y(u) and u=u(x)
2.4.2 [f(g(x)]' = f'(g(x))g'(x)
2.5 Finding 2nd Derivative
2.5.1 just take the derivative of the derivative
2.6 f(x) = e^x, f'(x) = e^x
2.6.1 f(x) = ln x, f'(x) =1/x
2.6.2 g(x) = loga(x), g'(x) = (1/ln a)(1/x)
2.6.2.1 f'(x) = ln(a)a^x
2.6.3 Chain Rule and Logs
2.6.3.1 g(x) = ln(f(x))
2.6.3.1.1 f'(x)/f(x)
2.6.4 ln(x) + 1/x (x')
3 Marginal Analysis/Linear Approximation
3.1 f(x+delta x) ~= f(x) + delta x(f'(x))
3.2 or Marginal Cost = C'(x)
4 Implicit Differentiation
4.1 Treat x as x and y as f(x), give derivative of y w/respect to x
4.2 Differentiate both sides w/respect to x with y as f(x)
4.2.1 Chain rule, differentiating terms with y
4.2.1.1 Solve for dy/dx in terms of x and y
5 Increasing/Decreasing Functions
5.1 Use 1st Derivative
5.1.1 Test points @ f'(x) = 0
5.1.1.1 Plug test points into f'(x)
5.1.1.2 And where f'(x) does not exist
5.1.1.2.1
5.1.1.3 CRITICAL POINTS!
5.2 Concavity
5.2.1 Use f''(x)
5.2.1.1 Test points @ f''(x) = 0
5.2.1.1.1 And where f''(x) does not exist
5.2.1.1.1.1 Inflection point where concavity changes
5.3 Positive/Negative
5.3.1 Test points on f(x)
6 Optimization
6.1 x = c is critical point if f'(c) = 0
6.1.1 x = c relative min/max if f'(c) changes sign
6.1.2 x = c abolute min/max
6.2 Closed Interval
6.2.1 find where f'(x)=0
6.2.1.1 plug those points into f(x)
6.2.1.1.1 compare against interval points for absolute max/min
6.3 Open Interval
6.3.1 find one critical point, where f'(x)=0
6.3.1.1 this shows concavity
7 Exponents
7.1 Compound interest
7.1.1 Yearly
7.1.1.1 P(t) = P(1+r)^t
7.1.2 Monthly
7.1.2.1 P(t) = P(1+r/12)^12t
7.1.3 Continuously
7.1.3.1 P(t) = Pe^rt

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