Chapter 4

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  1. 1. y'=y+t
  2. 2. Newton's Law of Cooling
    • y'=k(y-Te)
    • y=Te+Ae^(kt)
  3. 3. L'Hôpital's Rule
    • lim(f(x)/g(x))=lim(f'(x)/g'(x))
    • *If indeterminate forms infinity/infinity, 0/0
  4. 4. Limit Definition
    • lim(f(x))=L
    • x->a
    • Provided for any epsilon>0 there exists a delta>0 such that whenever |x-a|
  5. 5. Vertical asymptotes
    Denom=0, numerator doesn't
  6. 6. Horizontal asymptotes
    Degree of top <= degree of bottom
  7. 7. Oblique
    Degree of top is one more than bottom
  8. 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
  9. 9. To maximize, minimize
    Find critical points, endpoints.
  10. 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
  11. 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
  12. 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
  13. 13. Maclaurin polynomials
  14. 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
  15. 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
  16. 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]
  17. 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)
  18. 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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Chapter 4
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