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Beschreibung
Mindmap von GraceEChem, aktualisiert more than 1 year ago
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Erstellt von GraceEChem
vor fast 10 Jahre
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Calculus I
- Limits
- Infinite
limits
- Divide top and
bottom by
biggest power
- 1/x^n = 0
- Divide top and
bottom by
biggest power
- Finite
limits
- Continuous, plug
in constant
- 0/0 is
your
answer?
- Factor
and
Simplify
- Factor
and
Simplify
- 0/0 is
your
answer?
- Continuous, plug
in constant
- Infinite
limits
- Derivatives
- Product Rule
- (fg)' = f'g+g'f
- (fg)' = f'g+g'f
- Quotient Rule
- (f/g)' = (f'g - g'f) / g^2
- (f/g)' = (f'g - g'f) / g^2
- Power Rule for Functions
- (f^n)' = nf^(n-1)f'
- (f^n)' = nf^(n-1)f'
- Chain Rule
- dy/dx = (dy/du)(du/dx)
- for y=y(u) and u=u(x)
- for y=y(u) and u=u(x)
- [f(g(x)]' = f'(g(x))g'(x)
- dy/dx = (dy/du)(du/dx)
- Finding 2nd Derivative
- just take the
derivative of the
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
- f'(x) = ln(a)a^x
- Chain Rule
and Logs
- g(x) = ln(f(x))
- f'(x)/f(x)
- f'(x)/f(x)
- g(x) = ln(f(x))
- ln(x) + 1/x (x')
- f(x) = ln x,
f'(x) =1/x
- Product Rule
- Marginal
Analysis/Linear
Approximation
- f(x+delta x) ~= f(x) + delta x(f'(x))
- or Marginal Cost = C'(x)
- f(x+delta x) ~= f(x) + delta x(f'(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
- Solve for dy/dx in
terms of x and y
- Chain rule,
differentiating
terms with y
- Treat x as x and
y as f(x), give
derivative of y
w/respect to x
- 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!
- Plug test points into f'(x)
- Test points @ f'(x) = 0
- Concavity
- Use f''(x)
- Test points @ f''(x) = 0
- And where f''(x)
does not exist
- Inflection point
where
concavity
changes
- Inflection point
where
concavity
changes
- And where f''(x)
does not exist
- Test points @ f''(x) = 0
- Use f''(x)
- Positive/Negative
- Test points on f(x)
- Test points on f(x)
- Use 1st Derivative
- 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
- x = c relative
min/max if f'(c)
changes sign
- Closed Interval
- find where f'(x)=0
- plug those
points into
f(x)
- compare against
interval points for
absolute max/min
- compare against
interval points for
absolute max/min
- plug those
points into
f(x)
- find where f'(x)=0
- Open Interval
- find one
critical point,
where f'(x)=0
- this shows
concavity
- this shows
concavity
- find one
critical point,
where f'(x)=0
- x = c is critical point if f'(c) = 0
- Exponents
- Compound interest
- Yearly
- P(t) = P(1+r)^t
- P(t) = P(1+r)^t
- Monthly
- P(t) = P(1+r/12)^12t
- P(t) = P(1+r/12)^12t
- Continuously
- P(t) = Pe^rt
- P(t) = Pe^rt
- Yearly
- Compound interest
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