Limits Flashcards
1
Q
Explain limits
A
2
Q
Standard limits 1
A
3
Q
Standard limits 2
A
4
Q
Standard limits 3
A
5
Q
Standard limits 4
A
6
Q
Standard limits 5
A
7
Q
Standard limits 6
A
8
Q
Standard limits 7
A
9
Q
Standard limits 8
A
10
Q
Standard limits 9
A
11
Q
Standard limits 10
A
12
Q
Indeterminate forms (7)
A
13
Q
L-Hospital Rule
A
14
Q
If you have limit approaches to infinity and polynomial, what to do?
A
Take the highest degree common
15
Q
If you have 0x∞ form, what to to
A
If you have 0x∞ form, make it in 0/0 or ∞/∞ form
example: f(x).g(x) is in the 0x∞ form then write it as
16
Q
If you ∞-∞ indeterminate form, what to do
A
Write function into another form
for ex: make unrationalized into rationalized
17
Q
If x->0 then (1+x)ⁿ
A
If x->0 then (1+x)ⁿ ≅1+nx
18
Q
When will (1+x)ⁿ becomes 1+nx
A
when x->0
19
Q
Piecewise function
A
20
Q
How to break |f(x)|
A
21
Q
What is f(x) = |x|
A
22
Q
What is f(x) = |cos x|
A
23
Q
What to do in mode graph
A
Make the -ve y graph (i.e. graph in 3rd and 4th quadrant) +ve (i.e. make the mirror image w.r.t. to x axis of -ve)
24
Q
Greatest integer function
A
25
How to pronounce greatest integer function
Floor of x
26
Another name of gif
Step function
27
Notation of gif
28
Explain gif
(Short)
29
Explain continuity of gif
Discontinuous function at every integer
30
continuity of f(x)
31
if limit doesn't exit then
it means f(x) at x=a doesn't follow LHL = RHL = f(a)
and graph is discontinuous at x=a
32
Leibnitz rule
33
What is the differentiation of an integration with variable limits
34
Derivative of a function f at any point x
35
Differentiability of f(x) at x=a
36
Derivative by first principle
37
Explain relation between continuity and differentiability of f(x)
Every differentiable function is continuous but every continuous function need not be differentiable
For ex: f(x) = |x|
38
How to tell that function is differentiable or not (if graph is given)
39
Even function
f(-x) = f(x) ∀ x∈R
40
Odd function
f(-x) = -f(x) ∀ x∈R
41
Graph of even function
symmetric about y-axis
42
Graph of odd function
symmetric about origin
43
Differentiation of even function
is odd function
44
Differentiation of odd function
even function
45
Another name of derivative
gradient
46
Explain Partial differentiation
47
Total derivative of z
48
Partial derivative of z w.r.t x
49
Partial derivative of z w.r.t y
50
What is this (see pic)
51
What is this (see pic)
52
dz/dx
53
dz/dy
54
What is this(see pic)
55
another notation of this (see pic)
56
another notation of this (see pic)
57
What is this
58
What is this
59
Homogeneous function
u = f(x,y) is a homogeneous function of degree 'n' iff f(kx,ky) = kⁿf(x,y)
60
Shortcut of homogeneous function
Degree of every term is same
61
special point about homogeneous function
if you have rational function, both numerator and denominator are homogeneous then function is also homogeneous
62
degree of homogeneous function
degree(numerator) - degree(denominator)
63
64
65
66
67
68
Euler theorem types
Type 1: u = f(x,y) is a homogenous function of degree n
Type 2: u = f(x,y) + g(x,y) where f, g are homogeneous functions of degree n₁ and degree n₂ respectively
Type 3: Φ(u) = f(x,y) is a homogeneous function of degree 'n'
69
Euler theorem type 1
Type 1: u = f(x,y) is a homogenous function of degree n then
i.) xuₓ + yuᵧ = nu
ii.) x²uₓₓ + y²uᵧᵧ + 2xyuₓᵧ = n(n-1)u
70
Euler theorem type 2
Type 2: u = f(x,y) + g(x,y) where f, g are homogeneous functions of degree n₁ and degree n₂ respectively then
i.) xuₓ + yuᵧ = n₁f + n₂g
ii.) x²uₓₓ + y²uᵧᵧ + 2xyuₓᵧ = n₁(n₁-1)f + n₂(n₂-1)g
71
Euler theorem type 3
Type 3: Φ(u) = f(x,y) is a homogeneous function of degree 'n' then
i.) xuₓ + yuᵧ = nΦ(u)/Φ'(u) = F(u)
ii.) x²uₓₓ + y²uᵧᵧ + 2xyuₓᵧ = F(u) [F'(u) - 1]
72
What to do in this type of function where we need to apply homogeneous function concept
73
What is the relationship between
dy/dx and dx/dy ?
Both are reciprocals of each other
74
Chain rule
75
Mean value theorems (3)
1.) Roll's theorem
2.) Lagrange's mean value theorem
3.) Cauchy's mean value theorem
76
Rolle's theorem (brief)
77
Explain Rolle's theorem
78
What does f'(c) = 0 means if function satisfying the Rolle's theorem
If once the function is satisfying the Rolle's theorem, there exist at least one point where the function can reach its peak values like minimum values or maximum values.
79
Lagrange's mean value theorem (brief)
80
Explain Lagrange's mean value theorem
81
What does f'(c) gives in Lagrange's mean value theorem and what it doesn't give
Using Lagrange's mean value theorem, f'(c) gives derivative of f(c) at certain point b/w (a,b)
f'(c) doesn't gives maxima or minima since f'(c) is not zero
82
Cauchy’s mean value theorem
