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2. Newton's Law of Cooling
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3. L'Hôpital's Rule
- lim(f(x)/g(x))=lim(f'(x)/g'(x))
- *If indeterminate forms infinity/infinity, 0/0
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4. Limit Definition
- lim(f(x))=L
- x->a
- Provided for any epsilon>0 there exists a delta>0 such that whenever |x-a|
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5. Vertical asymptotes
Denom=0, numerator doesn't
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6. Horizontal asymptotes
Degree of top <= degree of bottom
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7. Oblique
Degree of top is one more than bottom
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8. Squeeze theorem
If for all x f(x)<=g(x)<=h(x), for x big and positive, if f(x)->L and h(x)->L, g(x)->L
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9. To maximize, minimize
Find critical points, endpoints.
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10. Related rates
- 1. Write an equation that 2+ quantities which vary with time
- 2. Differentiate both sides with respect to time (know rules)
- 3. The resulting equation relates the rates at which quantities vary
- 4. Solve for what is requested
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11. Newton's Method
Let f be any function and x0 any estimate to a root of f. For n>=1, the rule x(n+1)=xn-f(xn)/f'(xn) produces a sequence of estimates
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12. Taylor polynomial of order n at point x0 is:
- T(x)=f(xn)+f'(x0)(x-x0)+f''(x0)/2(x-x0)^2+...+f(n)(x0)/n!(x-x0)^n
- *Accurate around x0, better if more terms
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13. Maclaurin polynomials
X0=0
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14. Intermediate Value Theorem
The continuous function f on interval [a,b] takes on all range values between f(a) and f(b) at least once for domains between a and b
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15. Location Principle
For continuous function on interval [a,b] if f(a)<0 and f(b)>0, then there is at least one real c between a and b such that f(c)=0
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16. Extreme Value Theorem
If f is continuous on [a,b] then f assumes both a max and min value for some domain values in [a,b]
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17. Mean Value Theorem
If f is continuous on interval [a,b] and f is differentiable on (a,b), then there exists a c between a and b such that f'(c)=(f(b)-f(a))/(b-a)
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18. Rolle's Theorem
If f is continuous on (a,b) and differentiable on (a,b) and f(a)=f(b) then there exists a c between a and b such that f'(c)=0
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