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Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Document Description: Calculus of Single & Multiple Variables for Physics 2022 is part of Mathematical Models preparation. The notes and questions for Calculus of Single & Multiple Variables have been prepared according to the Physics exam syllabus. Information about Calculus of Single & Multiple Variables covers topics like Limits, Continuity, Differentiability, Partial Differentiation, Jacobian, Taylor’s Series and Maclaurine Series Expansion and Calculus of Single & Multiple Variables Example, for Physics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Calculus of Single & Multiple Variables.

Introduction of Calculus of Single & Multiple Variables in English is available as part of our Mathematical Models for Physics & Calculus of Single & Multiple Variables in Hindi for Mathematical Models course. Download more important topics related with notes, lectures and mock test series for Physics Exam by signing up for free. Physics: Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Table of contents
Limits
Continuity
Differentiability
Partial Differentiation
Jacobian
Taylor’s Series and Maclaurine Series Expansion
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Limits

The number A is said to be the limit of f (x) at x = a if for any arbitrary chosen positive

numbers, however small but not zero, there exists a corresponding number ε greater than zero such that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics    For all values of x for which 0 Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Right hand and Left hand Limit

If x approaches a from larger values of x than a, thenCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
"Put a + h for x in f(x) and make h approaches zero'’. In short, we have
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
If x approaches a from smaller values of x than a, then Calculus of Single & Multiple Variables Notes | Study Mathematical Models - PhysicsIn this case  

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
If both right hand and left hand limits of f(x), as x → a, exist and are equal in value, their common value, evidently, will be the limit of f(x), as x → a .

Theorem of Limits

If f (x) and g (x) are two functions then
1. Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
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L’Hospital’s Rule

If a function f (x) takes the form Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsthen we say that f (x) is indeterminate at x = a. If ∅(x) and ψ(x) are functions of x such that ∅(a) = 0 and ψ(a) = 0, then
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Some Important Standard Limits
1. Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
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Example 1: Find the limit Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 2: Find the value of Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Continuity

f (x) is said to be continuous at x = a if R = L = V i.e.,
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
In case a function is not defined at x = a , i.e. f(a) does not exist or in case

theR.H.L ≠ L.H.L , then we say that the function is discontinuous at x = a. Its graph will show a break at x = a.

Example:
(i)Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics is discontinuous at x = a as f (a) does not exist.
(ii) Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics is discontinuous at x = 0 because
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
(iii) f (x) = 1 - cos e1x is discontinuous at x = 0 as f (0) = 1 - cos e1/0 is undefined. It is oscillating.
Example: Let the function f (x) be defined by
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
f(x) will be continuous at those points only where rational and irrational values coincide i.e. ex = e1 - x ⇒ ex = Calculus of Single & Multiple Variables Notes | Study Mathematical Models - PhysicsSo /(x) is continuous at x = l/2 only.

Differentiability

f (x) is said to be differentiable at x = a if R' = L' i.e,
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example: f(x) = e-x2 is differentiable but g(x) = Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics is not differentiable.
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Right hand Limit Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Left hand Limit Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Thus R' ≠ L' means g(x)Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsis not differentiable.

Tangents and Normal

Let y = f (x) be a given curve and P(x, y)and Q(x + δx, y + δy) be two neighbouring points on it. Equation of the line PQ is
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - PhysicsCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
This line will be tangent to the given curve at P if Q → P which in tem means that δx → 0 and we know that lim Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Therefore the equation of the tangent is Y - y = Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Normal at (x, y)
The normal at (x, v) being perpendicular to tangent will have its slope as Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsand hence its equation is Y - y = Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Geometrical meaning of dy/dx
dv/dx represents the slope of the tangent to the given curve y = f(x) at any point (x, y)
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicswhere ψ is the angle which the tangent to the curve makes with +ve

direction of x-axis.
In case we are to find the tangent at any point (x1, y1) then Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsi.e. the value of  Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsat (x1, y1) will represent the slope of the tangent and hence its equation in this case will be
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Normal Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Condition for tangent to be parallel or perpendicular to x-axis

If tangent is parallel to x-axis or normal is perpendicular to x-axis thenCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
If tangent is perpendicular to x-axis or normal is parallel to x-axis then
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Maxima and Minima

For the function y = f (x) at the maximum as well as minimum point the tangent is parallel to x-axis so that its slope is zero.
Calculate Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics and solve for x. Suppose one root of Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsis at x = a .
If Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics= negative for x = a, then maximum at x = a .
If Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicspositive for x = a, then minimum at x = a .
If Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsat x = a , then find Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
IfCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics0 at x = a, neither maximum nor minimum at x = a.
If Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
If Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsi.e., positive at x = a, then y is minimum at x = a and if Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsi.e -ve at 

x = a , then y is maximum at x = a and so on.

Partial Differentiation

If a derivative of function of several independent variables be found with respect to any one of them, keeping the others as constants it is said to be partial derivative. The operation of finding the partial derivative of function of more than one independent variable is called partial differentiation.

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Euler Theorem of Homogeneous Function

If u is homogeneous function of x and y of degree n then Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
For n variable function f = f(x1,x2,x3.........xn) of degree n then
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Total derivative u = f(x,y) x = ∅(t) and y = ψ(t) then Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Differentiation of implicit functions f(x, y) = c be an implicit relation between x and y which defines as a differential function of x then
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Change in variable u = f(x,y), x = ∅(s,t) and y = ∅(s,t)
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Maxima and Minima (of function of two independent variables)

The necessary condition that f (x, y) should have maximum or minimum at x = a, y = b is that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Sufficient condition for maxima and minima
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Case 1: if f(x, y) will have maximum or minimum at x = a, y = b if rt >s2 further f(x, y) is maximum or minimum according as r in negative or positive.
Case 2: if f(x, y) will have maximum or minimum at x = a, y = b if rt < s2 further /(x, y) is saddle point.
Case 3: if rt = s2 this case is doubtful case and further advanced investigation is needed to determined whether f(x,y) a maximum or minimum at or not.

Example 3: For what values of x and y , does the integral Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsdt attain its maximum?

l(x,y) or f(x,y) =Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
fx = -(6-x-x2) = x2+x-6 = (x + 3)(x-2), fy =6 - y - y2 = -(y + 3)(y - 2)
f= 0 ⇒ x = 2,-3, f= 0 ⇒ y = 2,-3

so, stationary points are (2, 2),(2,-3),(-3,2)&(-3,-3)

fxx = 2x +1, fyy = -2y -1, fxy = 0

fxxfyy - (fxy)2  = - (2x +1) (2y +1)

At x = -3 and y = 2 and fxxfyy - (fxy)2 > 0

So maximum value of f (x, y) is obtained at x = -3 and y = 2

Example 4: If z = z(x, y) then prove that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 5: Find the first and second partial derivatives of z = x3 + y3 - 3axy and prove that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

We have z = x3 + y3 - 3axy
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
AlsoCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
We observe that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 6: If u = x2 Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Show that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics and Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

We have Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Similarly, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
and Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 7: If z = f (x + ct) + ∅(x - ct), prove that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

We have
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
and Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Again Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
and Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
From (i) and (ii), it follows that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
This is an important partial differential equation, known as wave equation.

Example 8: If u = log (x3 + y3 +z3 - 3xyz) , show that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

we have
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Now Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 9: If u = f (r) and x = r cos θ, y = r sin θ, prove that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

we have Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Similarly, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
now to find Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsetc,we write r = (x+ v2)1/2
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Similarly, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Substituting the values of Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsetc. in (i), we get
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 10: Show that Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics where Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics


Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
∴ z is a homogeneous function of degree 2 in x and y .
By Euler’s theorem, we getCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Hence (i) becomes
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics


Example 11: If z is a homogeneous function of degree n in x and y , show that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

By Euler's theorem, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(i)
Differentiating (i) partially w.r.t. x, we get Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(ii)
Again differentiating (i) partially w.r.t. y , we get Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(iii)

Multiplying (ii) by x and (iii) by y and adding, we get
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 12: Given u = sinCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsx = et and v = t2, find du/dt as a function of t. verify your dt result by direct substitution.

We have Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 13: If x increases at the rate of 2 cm /sec at the instant when x = 3 cm . and at what rate must y be changing in order that the function 2xy - 3x2y shall be neither increasing nor decreasing?

Let u = 2xy-3x2y, so that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

when x = 3 and y = 1, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsand u is neither increasing nor decreasing ,i.e.,Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
∴ (i) becomes 0 = (2-6x3)2 + (2x3-3x9)Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
or Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicscm/sec. Thus y is decresing at the rate of 32/21 cm/sec.

Example 14: If μ = F(x-y,y -z,z- x) prove that
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Put x-y = r,y-z = s and z-x = t, so that u = f (r,s,t) .
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(i)
Similarly, Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(ii)
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics(iii)
Adding (i), (ii) and (iii), we get the required result. 

Jacobian

If u and v are two function of two independent variable x and y then the determinant
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
is called the Jacobian of u and v with respect to x and y which is written as Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physicsor 

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
If u, v and w are functions of independent variable x, y and z then the determinant
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
is called the Jacobian of u , v and w with respect to of x, y z which is written as Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Properties of Jacobian

1. Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
2. Chain rule for Jacobian if u,v arc function of r,s are function of x,y then
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
3. If u1, u2, u3 instead of being given explicitly in terms x1, x2, x3 be connected with them equations such as
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
then Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
((-1)3 is for three variable system)
4. If u1, u2, u3 be functions of x1, x2, x3 then the necessary and sufficient condition for existence of a functional relationship of the form f1 (u1u2, u3) = 0 isCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Example 15: In a polar coordinates x = r cos θ y = r cos θ then findCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 16: Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 17: u = x2 - y2,v = 2xy x = rcosθ,y = rsinθ findCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Use the formulaCalculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 18: u = xyz,v = x2 + y2 + z2,w = x + y + z find Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

We are using property three
f1 = u - xyz, f= v - x2 - v- z2, f3 = w-x-y-z
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Taylor’s Series and Maclaurine Series Expansion

If a function f(x) has continuous derivatives up to (n + 1)th order, then this function can be expanded in the following way:
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
where Rn called the remainder after (n + 1) terms, is given by
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
When this expansion converges over a certain range of x that is,Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics then the expansion is called Taylor Series of f(x) expanded about a.
If a = 0, the series is called Maclaurin Series:
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Maclaurine’s Development

Changing a to 0 and h to x one will get
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Where fn (ξ) is identified as reminder term ξ = θx and 0 < θ < 1

Example 19: In the Taylor series expansion of exp x + sin(x) about x = π then what is coefficient of (x - π)2.

Taylor series about x = π
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

f(x) = exp x + sin(x)

f(π) = exp π + sin(π) = exp π

f' (x) = exp x + cos x ⇒ f'(π) = exp π + cos π = exp π -1
f'' (x) = expx - sinx ⇒ f''(π) = expπ  sinπ ⇒ f'' (π) = expπ
The coefficient of (x - π)2 is Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Example 20: Expand Maclaurin's series expand of tan x up to x3

f(x) = tanx ⇒ f(0) = 0
f'(x) = sec2 x ⇒ f'(0) = 1

f''(x) = 2 tan x sec2 x ⇒ f''(0) = 0

f''' (x) = 2sec2 x +6tan2 xsec2 x ⇒ f''' (0) = 2
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

Some Important Expansions
1. Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
2.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
3.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics 
4.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
5.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
6.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
7.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
8.Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
9. Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics


Example 21: Expand exp(sin x) by maclaurin series upto the term containing x4

Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics


Taylor 's theorem for function in two variable .
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Puth h = x - a ,k = y - b
f(x, y) = f(a, b) + [(x - a)fx (a, b) + (y - b)fy (a, 6)]
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics


Example 22: Expand ex log(1 + y) upto powers of x and y upto term of two degree.

f (x, y) = ex log(1 + y) ⇒ f (0,0) = 0
fx(x, y) = elog(1 + y) ⇒ fx (0,0) = 0,Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
fxx (x, y) = elog(1+y) ⇒ fxx  (0,0) = 0,Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables Notes | Study Mathematical Models - Physics

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