-
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)
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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(x1) < F(x2) whenever x1 < x2 in I.
(page 20)
-
Decreasing
F(x1) > F(x2) whenever F(x1) > F(x2) in I.
(Page 20)
-
d/dx (sin x) =
Cos x
(page 193)
-
d/dx (cos x) =
-sin x
(page 193)
-
d/dx (tan x) =
Sec2 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) =
-csc2 x
(page 193)
-
-
d/dx (xn) =
nxn-1
Page 187
-
-
-
(f+g) ' =
f ' + g '
Page 187
-
(f-g) ' =
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)(un) = (nun-1)(du/dx)
- Alternatively
- (d/dx)(g(x))n = n(g(x))n-1 •g '(x)
-
d/dx(sinh-1(x)) =
Chapter 3
1 / ((1+x2)1/2)
-
d/dx(cosh-1(x)) =
Chapter 3
1 / ((x2-1)1/2)
-
d/dx((tanh-1(x)) =
Chapter 3
1 / (1-x2)
-
d/dx(csch-1(x)) =
Chapter 3
-1 / (|x|(x2+1)1/2)
-
d/dx(sech-1(x)) =
Chapter 3
-1 / (x(1-x2)1/2)
-
d/dx(coth-1(x)) =
Chapter 3
1 / (1-x2)
-
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].
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