Application Of Derivatives Class 12 Notes- Chapter 6

What are Derivatives?

Derivatives are referred to the situation when a quantity p varies with respect to another quantity q, fulfilling a condition p =f (q) and dp/dq (or f ‘ (q)) represents the change of rate of p with respect to q and dp/dq where q = \(q_{0}\) or f ‘ (\(q_{0}\)) represents the rate of change of p with respect to q at q = \(q_{0}\).

Chain Rule

Let us consider two variables p and q which vary with respect to the presence of another variable r, which means p = f(r) and q = g(r), then according to the chain rule,

\(\frac{\partial q}{\partial p} = \frac{\partial q}{\partial r}/\frac{\partial p}{\partial r}, if \frac{\partial p}{\partial r}\neq 0\)

Increasing and Decreasing Function

A Function p can be said increasing when:

  • We can see an increase on an interval (a,b) if \(x_{1}<x_{2}\) in (a,b) =>\(p(x_{1})<p(x_{2})\) for all \(x_{1}, x_{2}\epsilon (a,b)\)

A function is said to be decreasing when:

  • if \(x_{1}<x_{2}\) in (a,b) when: \(x_{1}>x_{2}\) for all \(x_{1}, x_{2}\epsilon (a,b)\)

Constant

The constant term in (a,b) if p(x)= q for all x \epsilon (a,b)\) and q is a constant

Equation of Tangent

The equation of a tangent is given at a point \((x_{0}, y_{0})\) to the curve y =f(x) is representated by
y-yo=dy/dx](xo,yo)(x-xo)

Critical point

The point q inside the domain of a function f at which f ‘ (q) =0 or f cannot be differentiable is referred to as the critical point of f.

 

Important Questions

  1. Find out the value of m if line y=mx+1 which is a tangent to the curve \(y^{2}=4x\).
  2. For the curve \(2y+x^{2}=3\) is ?
  1. A semicircular opening is surmounted on a window which is in the form of a rectangle The total perimeter of the window is 50 m. Find the dimensions of the window to admit maximum light through the whole opening.
  1. A point on the hypotenuse of a triangle is at distance p and q from the sides of the triangle.
  1. For what values of a the function f given by f(x) = \(x^{2}\) + ax + 1 is increasing On [1, 2]?

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