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1769425
Calculus I
Description
Mind Map on Calculus I, created by GraceEChem on 08/12/2014.
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math 111 - university calculus i
calculus 1
Mind Map by
GraceEChem
, updated more than 1 year ago
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Created by
GraceEChem
almost 10 years ago
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Resource summary
Calculus I
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
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')
Marginal Analysis/Linear Approximation
f(x+delta x) ~= f(x) + delta x(f'(x))
or Marginal Cost = C'(x)
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
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)
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
Exponents
Compound interest
Yearly
P(t) = P(1+r)^t
Monthly
P(t) = P(1+r/12)^12t
Continuously
P(t) = Pe^rt
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