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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)

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".

Increasing
F(x_{1}) < F(x_{2}) whenever x_{1} < x_{2} in I.
(page 20)

Decreasing
F(x_{1}) > F(x_{2}) whenever F(x_{1}) > F(x_{2}) in I.
(Page 20)

d/dx (sin x) =
Cos x
(page 193)

d/dx (cos x) =
sin x
(page 193)

d/dx (tan x) =
Sec^{2} x
(page 193)

d/dx (csc x) =
csc x cot x
(page 193)

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

d/dx (cot x) =
csc^{2} x
(page 193)


d/dx (x^{n}) =
nx^{n1}
Page 187

d/dx (e^{x}) =
e^{x}
Page 187


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

(fg) ' =
f '  g '
Page 187

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


Lim sin x/ x =
X>0
1
Page 190

Lim (cos x 1)/x =
x>0
0
Page 192

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

Power Rule combined with the Chain Rule
If "n" is any real number and u = g(x) is differentiable, then
 (d/dx)(u^{n}) = (nu^{n1})(du/dx)
 Alternatively
 (d/dx)(g(x))^{n} = n(g(x))^{n1} •g '(x)

d/dx(sinh^{1}(x)) =
Chapter 3
1 / ((1+x^{2})^{1/2})

d/dx(cosh^{1}(x)) =
Chapter 3
1 / ((x^{2}1)^{1/2})

d/dx((tanh^{1}(x)) =
Chapter 3
1 / (1x^{2})

d/dx(csch^{1}(x)) =
Chapter 3
1 / (x(x^{2}+1)^{1/2})

d/dx(sech^{1}(x)) =
Chapter 3
1 / (x(1x^{2})^{1/2})

d/dx(coth^{1}(x)) =
Chapter 3
1 / (1x^{2})

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.

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.

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.

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].

