calculus Flashcards
(104 cards)
1
Q
continuity
A
no jumps at x∈D
2
Q
smoothness
A
has a tangent at x∈D
3
Q
f(x) is smooth at x => continuous?
A
continuous at x
4
Q
f(x) is continuous at x => smooth?
A
not necessarily
5
Q
limit def
A
the value that a function approaches as the argument approaches some value
6
Q
when does a limit not exist
A
left limit ≠ right limit
7
Q
f(x) is continuous at a => lim(x->a) f(x) = ?
A
f(a)
8
Q
f(x) has a horizontal asymptote at y=a => lim(x->∞) f(x) = ?
A
a … constant
9
Q
lim(x->a) b = ?
a is a real number or +/- ∞, b is a constant
A
b
limit of a constant is constant
10
Q
lim(x->a) b(f(x)) = ?
a is a real number or +/- ∞, b is a constant
A
b lim(x->a) f(x)
11
Q
lim(x->a) (f(x) ± g(x)) =
a is a real number or +/- ∞, b is a constant
A
lim(x->a) f(x) ± lim(x->a) g(x)
12
Q
lim(x->a) (f(x) x g(x)) =
a is a real number or +/- ∞, b is a constant
A
lim(x->a) f(x) x lim(x->a) g(x)
13
Q
lim(x->a) (f(x) / g(x)) =
a is a real number or +/- ∞, b is a constant
A
lim(x->a) f(x) / lim(x->a) g(x)
14
Q
0/0 =
A
indeterminate form
15
Q
∞ / ∞ =
A
indeterminate form
16
Q
∞ - ∞ =
A
indeterminate form
17
Q
1^∞ =
A
indeterminate form
18
Q
∞^0
A
indeterminate form
19
Q
0^0 =
A
indeterminate form
20
Q
0 x ∞ =
A
indeterminate form
21
Q
∞ + ∞ =
A
∞
22
Q
∞^∞ =
A
∞
23
Q
∞^(-∞) =
A
0
24
Q
0^∞ =
A
0
25
derivative def
gradient of the curve/tangent to a curve
rate of change of f(x) at x = x1
26
differentation by first principle
| y = kx + b ... tangent
k = lim (h->0) (f(x+h) - f(x))/h = f'(x)
27
the normal
line perpendicular to the tangent
28
c' = ?
| c is a constant
0
29
(ax + b)' = ?
a
30
(x^n)' = ?
nx^(n-1)
31
((c)(f(x)))' = ?
| c is a constant
cf'(x)
32
(g(x) ± f(x))' = ?
g'(x) ± f'(x)
33
(f(x) x g(x)) = ?
(f'(x) x g(x)) + (f(x) x g'(x))
34
(f(x) / g(x))' = ?
((f'(x) x g(x)) - (f(x) x g'(x))/(g^2(x))
35
f(g(x))' = ?
f'(g(x)) x g'(x)
inside function into outside differentiated -> multiply by inside'
36
turning point
| def, when
the function changes from increasing to decreasing or vice versa
local extremum
turning point => f'(x) = 0; not <=>
37
lim(x->0) (cosx-1)/x = ?
0
38
lim(x->0) (sinx/x) =
1
39
sin'x = ?
cosx
40
cos'x = ?
-sinx
41
stationary point
| condition, types
the tangent is horizontal => f'(x) = 0
local extrema (max/min), inflection point
42
f'(x) > 0
f(x) is increasing at x
43
f'(x) < 0
f(x) is decreasing at x
44
f'(x) = 0
f(x) is neither increasing nor decreasing (stationary point)
45
f''(x) > 0
f(x) concave-up at x
46
f''(x) < 0
f(x) is concave-down at x
47
f'(x) = 0 and f''(x) > 0
local min at x
48
f'(x) = 0 and f''(x) < 0
local max at x
49
inflection point on f(x) => ? on f'(x)
local maxima/minima at x
50
odd f(x) => ? f'(x)
even f'(x)
51
even f(x) => ? f'(x)
odd f'(x)
52
f(x) is steeper at x => f'(x) ?
f'(x) is greater (closer to infinity)
53
inflection point at x on f(x) < = > ? f'(x) < = > ? f''(x)
turning point on f'(x) at x
0 at f''(x) at x
54
finding inflection points
sign chart of f''(x)
sign change => inflection point
55
concave-up
| tangents, line segments, f''(x)
tangents are below the graph
line segments are above the graph
f''(x) > 0
56
concave-down
| tangents, line segments, f''(x)
tangents are above the graph
line segments are below the graph
f''(x) < 0
57
lim(x→∞) sinx
does not exist
58
tan’x = ?
sec2x = 1/cos2x
59
cot’x = ?
csc2x = 1/sin2x
60
sec’x = ?
secx tanx
61
csc’x = ?
-cscx cotx
62
arctan’x = ?
1/(1+x2)
63
arcsin’x = ?
1/√(1-x2)
64
arccos’x = ?
-1/√(1-x2)
65
(f-1(x))’ = ?
| derivative of an inverse function
1/(f’(f-1(x)))
66
(log(a)x)’ = ?
1/xlna
67
(a^x)’ = ?
a^x * lna
68
ln'x = ?
1/x
69
(e^x)’
e^x
70
function which differentiates into itself
multiples of e^x
71
when is a derivative undefined?
graph is vertical
72
l'hopital's rule
lim(x→c) f(x)/g(x) = lim(x→c) f’(x)/g’(x)
73
conditions for l'hopital
* f and g are smooth
* g’(x) ≠ 0 for x ≠ c
* inserting directly leads to an indeterminate form
* lim(x→c) f(x)/g(x) exists (a constant, ∞, -∞)
* lim(x→c) f(x) = 0, lim(x→c) g(x) = 0 OR lim(x→c) f(x) = ∞ or -∞, lim(x→c) g(x) = ∞ or -∞
74
average velocity
v = (x1+x2+...+xn)/(t1+t2+...+tn)
v changes linearly ⇒ average v = (v1 + v2)/2
75
average acceleration
a = (v1+v2+...+vn)/(t1+t2+...+tn)
a changes linearly ⇒ average a = (a1 + a2)/2
76
accelerating => speed, a and v
⇒ increasing speed ⇒ a and v have the same sign
77
deccelerating => speed, a and v
⇒ decreasing speed ⇒ a and v have opposite signs
78
sign of v
| =0, >0, < 0, changes sign
v = 0 ⇒ at rest
v > 0 ⇒ moving to the right
v < 0 ⇒ moving to the left
changes sign ⇒ changes direction
79
sign of a
| > 0, < 0
a > 0 ⇒ accelerating to the right, deccelerating to the left (velocity is increasing)
a < 0 ⇒ accelerating to the left, deccelerating to the right (velocity is decreasing)
80
indefinite integral
F(x) = ∫f(x)dx ⇒ a family of functions which differ by a constant (F(x) + C = G(x))
81
∫0dx = ?
c
82
∫dx
x + c
83
∫kdx
kx + c
84
∫x^ndx
x^(n+1)/(n+1) + c; n≠1
85
∫x^(-1)dx
ln|x|+ c
86
∫cosxdx
sinx + c
87
∫sinxdx
-cosx + c
88
∫(secx)^2dx
tanx + c
89
∫a^xdx
a^x/lna + c
90
∫e^xdx
e^x + c
91
∫log(a)xdx
(x/lna)(lnx - 1) + c
92
∫(1/√(1-x^2))dx
arcsinx + c
93
∫-(1/√(1-x^2))dx
arccosx + c = -arcsinx + c
94
∫1/(1+x^2)dx
arctanx + c
95
∫tanx secx dx
secx + c
96
∫cotx cscx dx
-cscx dx
97
∫(cscx)^2 dx
-cotx + c
98
definite integral
∫(a, b)f(x)dx = area under a curve (Riemonn sum)
99
-∫(b, a)f(x)dx = ?
∫(a, b)f(x)dx
100
∫(a, b)f(x)dx + ∫(b, c)f(x)dx
∫(a, c)f(x)dx
101
∫(a,a)f(x)dx =
0
102
∫(a,b)cdx =
c(b-a)
103
average value of a function
av. f(x) = ∫abf(x)dx / (b – a)
104
first fundamental theorem of calculus
∫(a,b)f(x)dx = F(b) - F(a)
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