Find \(\frac{dy}{dx}\),if y=12(1-cost),x=10(t-sint),\(-\frac{\pi}{2}\)<t<\(\frac{\pi}{2}\)
Find \(\frac{dy}{dx}\),if y=12(1-cost),x=10(t-sint),\(-\frac{\pi}{2}\)<t<\(\frac{\pi}{2}\)
Solution and Explanation
It is given that ,y=12(1-cost),x=10(t-sint)
∴\(\frac{dy}{dx}\)=\(\frac{d}{dt}\)(10(t-sint))=10.\(\frac{d}{dt}\)(t-sint)=10(1-cost)
\(\frac{dy}{dt}\)=\(\frac{d}{dt}\)[12(1-cost)]=12.\(\frac{d}{dt}\)(1-cost)=12.[0-(-sint)]=12sint
∴\(\frac{dy}{dt}\)=\(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) =\(\frac{12sin\,t}{10(1-cos\,t)}\)
=\(\frac{12.2sin\frac{t}{2}cos\frac{t}{2}}{10.2sin\frac{2t}{2}}\)=\(\frac{6}{5}cot\frac{t}{2}\)
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Notes on Continuity and differentiability
Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.