# Calculus 1A, College of the Desert, Chapters1-4.txt

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1. ﻿Lim F(x) = L
x-->a

• We say "the limit of F(x), as "x" approaches "a", equals "L".
• If we can make the values of F(x) arbitrarily close to "L" (as close to "L" as we like) by taking "x" to be sufficiently vlose to "a" ( on either side of "a") but not equal to "a".

(page 88)
2. Function
A function "F" is a rule that assigns to each element "x" in a set "D" exactly one element, called F(x), in a set "E".
3. Increasing
F(x1) < F(x2) whenever x1 < x2 in I.

(page 20)
4. Decreasing
F(x1) > F(x2) whenever F(x1) > F(x2) in I.

(Page 20)
5. d/dx (sin x) =
Cos x

(page 193)
6. d/dx (cos x) =
-sin x

(page 193)
7. d/dx (tan x) =
Sec2 x

(page 193)
8. d/dx (csc x) =
-csc x cot x

(page 193)
9. d/dx (sec x) =
Sec x tan x

(page 193)
10. d/dx (cot x) =
-csc2 x

(page 193)
11. d/dx (C) =
0

(page 187)
12. d/dx (xn) =
nxn-1

Page 187
13. d/dx (ex) =
ex

Page 187
14. (cf)' =
cf '

Page 187
15. (f+g) ' =
f ' + g '

Page 187
16. (f-g) ' =
f ' - g '

Page 187
17. (fg) ' =
fg ' + f 'g

Page 187
18. (f/g) ' =
• (gf ' - fg ')
• g2

Page 187
19. Lim sin x/ x =
X-->0
1

Page 190
20. Lim (cos x -1)/x =
x-->0
0

Page 192
21. Chain Rule
• If "g" is differentiable at "x" and "f" is differentiable at g(x), then the composite function F=f • g defined by F(x) = f(g(x)) is differentiable at "x" and F ' is given by the product
• F '(x) = f '(g(x)) • g '(x)
• In Leibniz notation' if y = f(u) and u = g(x) are both differentiable functions, then
• dy/dx = (dy/du)(du/dx)

Page 197
22. Power Rule combined with the Chain Rule
If "n" is any real number and u = g(x) is differentiable, then

• (d/dx)(un) = (nun-1)(du/dx)
• Alternatively
• (d/dx)(g(x))n = n(g(x))n-1 •g '(x)
23. d/dx(sinh-1(x)) =
Chapter 3
1 / ((1+x2)1/2)
24. d/dx(cosh-1(x)) =
Chapter 3
1 / ((x2-1)1/2)
25. d/dx((tanh-1(x)) =
Chapter 3
1 / (1-x2)
26. d/dx(csch-1(x)) =
Chapter 3
-1 / (|x|(x2+1)1/2)
27. d/dx(sech-1(x)) =
Chapter 3
-1 / (x(1-x2)1/2)
28. d/dx(coth-1(x)) =
Chapter 3
1 / (1-x2)
29. Absolute maximum
• aka: global maximum
• A function "F" has an absolute or global maximum at "c" if F(c) >_ F(x) for all x in D, where D is the domain of F. The number F(c) is called the Maximum Value of F on D. Also called an extreme value.
30. Absalute Minimum
F has an absalute minimum at "c" if F(c) <_F(x) for all x in D and the number F(c) is called Minimum Value of F on D. Also called an extreme value.
31. Local Maximum
A function F has a Local Maximum (or relitive maximum) at "c" if F(c) >_F(x) when x is near c. [This means that F(c) >_ F(x) for all x in some open interval containing c]. Similary, F has a Local Minimum at c if F(c) <_ F(x) when x is near c.
32. Extreme Value Theorem
If F is continuouse on a closed interval [a,b], then F attains an absolute maximum value F(c) and an absalute mimimum value F(d) at some numbers c and d in [a,b].
 Author: Mattyj1388 ID: 33033 Card Set: Calculus 1A, College of the Desert, Chapters1-4.txt Updated: 2011-06-23 22:27:33 Tags: Chapters1 Calculus 1A Folders: Description: general info Show Answers: