# Chapter 4

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1. 1. y'=y+t
y=Ce^t-t-1
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
X0=0
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
 Author: SamboTheMan ID: 292169 Card Set: Chapter 4 Updated: 2014-12-28 22:51:27 Tags: Calculus Folders: Description: APC Show Answers: