Maths [Radius of curvature; partial diff]

Question 1

derivate angle between radius vector and tangent

Answer 1

do doo

Question 2

what is the angle between two polar curves

Answer 2

Φ1-Φ2 mod
if tan(Φ1)xtan(Φ2)=-1 then the angle is 90 degrees

Question 3

tanΦ in terms of radius

Answer 3

r dtheta/dr

Question 4

what is radius of curvature

Answer 4

y, a change in Ψ represents the bending of the curve C and the ratio dΨ/ds represents
the ratio of bending this rate of bending is called the curvature of the curves denoted by K or kappa
P (ro)= 1/K

Question 4

Radius of curvature in Cartesian form

Answer 4

P = [1+ (y’)^2]^3/2 whole divided by y’’
if We note that when y’=infinte the
use P = [1+ (dx/dy)^2]^3/2 whole divided by dx/dy

Question 5

Radius of Curvature in Polar Curve r=f(θ)

Answer 5

P= {r^2+r1^2}^3/2 whole divided by r^2 +2 r1^2 - rr2

Question 6

Centre of curvature formula

Answer 6

The Centre of Curvature at a point „P‟ of a curvature is the point “C” which lies on the Positive direction of the normal at „P‟ and is at a distance (in magnitude) from it.
(X,Y)= ( x- y’(1+y’^2)/y’’ , y +(1+y’^2)/y’’)

Question 7

Circle of curvature

Answer 7

The circle of curvature at a point „P‟ of a curve is the circle whose centre is at the centre of Curvature C and whose radius is in magnitude.

Question 8

Taylor’s Mean Value Theorem and formula

Answer 8

Suppose a function satisfies the following two conditions:
(i) and it’s first (n-1) derivatives are continuous in a closed interval [a,b]
(ii) is differentiable in the open interval (a,b)
Then there exists at least one point c in the open interval such that
f(x)= f(a)+(x-a)f’(a)+(x-a)^2f’‘(a)/2!+(x-a)^3f’’‘(a)/3! + (x-a)^4f’’’‘(a)/4! +……….

Question 8

Maclaurin’s series

Answer 8

f(x)= f(0) + xf’(0)+ x^2 f’‘(0)/2! + x^3 f’’‘(0)/3!