Limits Flashcards
Explain limits
Standard limits 1
Standard limits 2
Standard limits 3
Standard limits 4
Standard limits 5
Standard limits 6
Standard limits 7
Standard limits 8
Standard limits 9
Standard limits 10
Indeterminate forms (7)
L-Hospital Rule
If you have limit approaches to infinity and polynomial, what to do?
Take the highest degree common
If you have 0x∞ form, what to to
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
If you ∞-∞ indeterminate form, what to do
Write function into another form
for ex: make unrationalized into rationalized
If x->0 then (1+x)ⁿ
If x->0 then (1+x)ⁿ ≅1+nx
When will (1+x)ⁿ becomes 1+nx
when x->0
Piecewise function
How to break |f(x)|
What is f(x) = |x|
What is f(x) = |cos x|
What to do in mode graph
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)
Greatest integer function
How to pronounce greatest integer function
Floor of x
Another name of gif
Step function
Notation of gif
Explain gif
(Short)
Explain continuity of gif
Discontinuous function at every integer
continuity of f(x)
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
Leibnitz rule
What is the differentiation of an integration with variable limits
Derivative of a function f at any point x
Differentiability of f(x) at x=a
Derivative by first principle
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|
How to tell that function is differentiable or not (if graph is given)
Even function
f(-x) = f(x) ∀ x∈R
Odd function
f(-x) = -f(x) ∀ x∈R
Graph of even function
symmetric about y-axis
Graph of odd function
symmetric about origin
Differentiation of even function
is odd function
Differentiation of odd function
even function
Another name of derivative
gradient
Explain Partial differentiation
Total derivative of z
Partial derivative of z w.r.t x
Partial derivative of z w.r.t y
What is this (see pic)
What is this (see pic)
dz/dx
dz/dy
What is this(see pic)
another notation of this (see pic)
another notation of this (see pic)
What is this
What is this
Homogeneous function
u = f(x,y) is a homogeneous function of degree ‘n’ iff f(kx,ky) = kⁿf(x,y)
Shortcut of homogeneous function
Degree of every term is same
special point about homogeneous function
if you have rational function, both numerator and denominator are homogeneous then function is also homogeneous
degree of homogeneous function
degree(numerator) - degree(denominator)
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’
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
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
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]
What to do in this type of function where we need to apply homogeneous function concept
What is the relationship between
dy/dx and dx/dy ?
Both are reciprocals of each other
Chain rule
Mean value theorems (3)
1.) Roll’s theorem
2.) Lagrange’s mean value theorem
3.) Cauchy’s mean value theorem
Rolle’s theorem (brief)
Explain Rolle’s theorem
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.
Lagrange’s mean value theorem (brief)
Explain Lagrange’s mean value theorem
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
Cauchy’s mean value theorem
