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
For all values of x for which 0
If x approaches a from larger values of x than a, then
"Put a + h for x in f(x) and make h approaches zero'’. In short, we have
If x approaches a from smaller values of x than a, then In this case
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 .
If f (x) and g (x) are two functions then
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If a function f (x) takes the form then 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
Some Important Standard Limits
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Example 1: Find the limit
Example 2: Find the value of
f (x) is said to be continuous at x = a if R = L = V i.e.,
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) is discontinuous at x = a as f (a) does not exist.
(ii) is discontinuous at x = 0 because
(iii) f (x) = 1  cos e^{1x} is discontinuous at x = 0 as f (0) = 1  cos e^{1/0} is undefined. It is oscillating.
Example: Let the function f (x) be defined by
f(x) will be continuous at those points only where rational and irrational values coincide i.e. e^{x} = e^{1  x} ⇒ e^{x} = So /(x) is continuous at x = l/2 only.
f (x) is said to be differentiable at x = a if R' = L' i.e,
Example: f(x) = e^{x2} is differentiable but g(x) = is not differentiable.
Right hand Limit
Left hand Limit
Thus R' ≠ L' means g(x)is 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
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
Therefore the equation of the tangent is Y  y =
Normal at (x, y)
The normal at (x, v) being perpendicular to tangent will have its slope as and hence its equation is Y  y =
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)
∴ where ψ is the angle which the tangent to the curve makes with +ve
direction of xaxis.
In case we are to find the tangent at any point (x_{1}, y_{1}) then i.e. the value of at (x_{1}, y_{1}) will represent the slope of the tangent and hence its equation in this case will be
Normal
If tangent is parallel to xaxis or normal is perpendicular to xaxis then
If tangent is perpendicular to xaxis or normal is parallel to xaxis then
For the function y = f (x) at the maximum as well as minimum point the tangent is parallel to xaxis so that its slope is zero.
Calculate and solve for x. Suppose one root of is at x = a .
If = negative for x = a, then maximum at x = a .
If positive for x = a, then minimum at x = a .
If at x = a , then find
If0 at x = a, neither maximum nor minimum at x = a.
If
If i.e., positive at x = a, then y is minimum at x = a and if i.e ve at
x = a , then y is maximum at x = a and so on.
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.
If u is homogeneous function of x and y of degree n then
For n variable function f = f(x_{1},x_{2},x_{3}.........x_{n}) of degree n then
Total derivative u = f(x,y) x = ∅(t) and y = ψ(t) then
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
Change in variable u = f(x,y), x = ∅(s,t) and y = ∅(s,t)
The necessary condition that f (x, y) should have maximum or minimum at x = a, y = b is that
Sufficient condition for maxima and minima
Case 1: if f(x, y) will have maximum or minimum at x = a, y = b if rt >s^{2} 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 < s^{2} further /(x, y) is saddle point.
Case 3: if rt = s^{2} 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 dt attain its maximum?
l(x,y) or f(x,y) =
f_{x} = (6xx^{2}) = x^{2}+x6 = (x + 3)(x2), f_{y} =6  y  y^{2} = (y + 3)(y  2)
f_{x }= 0 ⇒ x = 2,3, f_{y }= 0 ⇒ y = 2,3so, stationary points are (2, 2),(2,3),(3,2)&(3,3)
f_{xx }= 2x +1, f_{yy} = 2y 1, f_{xy} = 0
f_{xx}f_{yy } (f_{xy})^{2} =  (2x +1) (2y +1)
At x = 3 and y = 2 and f_{xx}f_{yy } (f_{xy})^{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
Example 5: Find the first and second partial derivatives of z = x^{3} + y^{3}  3axy and prove that
We have z = x^{3} + y^{3}  3axy
Also
We observe that
Example 6: If u = x^{2}
Show that and
We have
∴Similarly,
and
Example 7: If z = f (x + ct) + ∅(x  ct), prove that
We have
and
Again
and
From (i) and (ii), it follows that
This is an important partial differential equation, known as wave equation.
Example 8: If u = log (x^{3} + y^{3} +z^{3}  3xyz) , show that
we have
Now
Example 9: If u = f (r) and x = r cos θ, y = r sin θ, prove that
we have
Similarly,
∴
now to find etc,we write r = (x^{2 }+ v^{2})^{1/2}
Similarly,
Substituting the values of etc. in (i), we get
Example 10: Show that where
∴ z is a homogeneous function of degree 2 in x and y .
By Euler’s theorem, we get
Hence (i) becomes
Example 11: If z is a homogeneous function of degree n in x and y , show that
By Euler's theorem, (i)
Differentiating (i) partially w.r.t. x, we get
(ii)
Again differentiating (i) partially w.r.t. y , we get
(iii)Multiplying (ii) by x and (iii) by y and adding, we get
Example 12: Given u = sinx = e^{t} and v = t^{2}, find du/dt as a function of t. verify your dt result by direct substitution.
We have
∴
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  3x^{2}y shall be neither increasing nor decreasing?
Let u = 2xy3x^{2}_{y}, so that
when x = 3 and y = 1, and u is neither increasing nor decreasing ,i.e.,
∴ (i) becomes 0 = (26x3)2 + (2x33x9)
or cm/sec. Thus y is decresing at the rate of 32/21 cm/sec.
Example 14: If μ = F(xy,y z,z x) prove that
Put xy = r,yz = s and zx = t, so that u = f (r,s,t) .
(i)
Similarly, (ii)
(iii)
Adding (i), (ii) and (iii), we get the required result.
If u and v are two function of two independent variable x and y then the determinant
is called the Jacobian of u and v with respect to x and y which is written as or
If u, v and w are functions of independent variable x, y and z then the determinant
is called the Jacobian of u , v and w with respect to of x, y z which is written as
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2. Chain rule for Jacobian if u,v arc function of r,s are function of x,y then
3. If u1, u_{2}, u_{3} instead of being given explicitly in terms x1, x2, x3 be connected with them equations such as
then
((1)^{3} is for three variable system)
4. If u_{1}, u_{2}, u_{3} be functions of x_{1}, x_{2}, x_{3} then the necessary and sufficient condition for existence of a functional relationship of the form f1 (u_{1}, u_{2}, u_{3}) = 0 is
Example 15: In a polar coordinates x = r cos θ y = r cos θ then find
Example 16:
Example 17: u = x^{2}  y^{2},v = 2xy x = rcosθ,y = rsinθ find
Use the formula
Example 18: u = xyz,v = x^{2} + y^{2} + z^{2},w = x + y + z find
We are using property three
f_{1} = u  xyz, f_{2 }= v  x^{2}  v^{2 } z^{2}, f_{3} = wxyz
If a function f is such that f^{n1} is continuous in [a, a + h]and derivative in ]a, a + h[ then their exist a number θ∈ ]0, l[ such that
One can also write Taylor expansion by putting a + h = x
where f^{n} (ξ) is identified as reminder term ξ = a + θ(x  a) and 0 < θ < 1
The (n + 1)th term of the expansion is
Changing a to 0 and h to x one will get
Where f^{n} (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
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
Example 20: Expand Maclaurin's series expand of tan x up to x^{3}
f(x) = tanx ⇒ f(0) = 0
f'(x) = sec^{2} x ⇒ f'(0) = 1f''(x) = 2 tan x sec^{2} x ⇒ f''(0) = 0
f''' (x) = 2sec^{2} x +6tan^{2} xsec^{2} x ⇒ f''' (0) = 2
Some Important Expansions
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Example 21: Expand exp(sin x) by maclaurin series upto the term containing x^{4}
Taylor 's theorem for function in two variable .
Puth h = x  a ,k = y  b
f(x, y) = f(a, b) + [(x  a)f_{x} (a, b) + (y  b)f_{y} (a, 6)]
Example 22: Expand e^{x} log(1 + y) upto powers of x and y upto term of two degree.
f (x, y) = e^{x} log(1 + y) ⇒ f (0,0) = 0
f_{x}(x, y) = e^{x }log(1 + y) ⇒ f_{x} (0,0) = 0,
f_{xx} (x, y) = e^{x }log(1+y) ⇒ f_{xx} (0,0) = 0,
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