Application of Derivatives Summary

Let y = f(x) be a smooth curve and P(x,y) be a point on the curve.

• Equation of the tangent at (x₁, y₁) in the curve y = f (x) is

y - y₁

• Equation of the normal at (x₁, y₁) in the curve y = f (x) is

• If m = 0 the tangent at (x₁, y₁) is parallel to x-axis.

• Angle of intersection of the curves is the acute angle between their tangents at the point of intersection.

Let y = f₁(x) and y = f₂(x) be two curves intersecting at P.

• If m₁ = m₂ the curves touch each other.

• If m₁m₂ = -1 the curves are orthogonal.

• The condition for the function f(x) to be increasing at x = a if

f^'(a) >0.

• The condition for f(x) to be decreasing at x = a if f^'(a) < 0.

A function f(x) is said to be strictly increasing at x = a if f(x) > f(a) whenever x > a in the neighbour hood of a.

A function f(x) is said to be strictly decreasing at x = a if f(x) < f(a) whenever x>a in the neighbour hood of a

• If f^'(a) = 0 then x = a is called a stationary point.

If f^'(a)=0, x=a is called the stationary point and the tangent at (a_,f(a)) is parallel to x - axis.

• A function f(x) is said to have a maximum at x = a if

• The conditions for y = f(x) to have a maximum at x = a are

• A function f(x) is said to have a minimum at x = a

• The conditions for y = f(x) to have a minimum at x = a are

x = a is called a point of inflexion.

• Rolle's theorem: If a function f(x) is such that

(i) f (x) is continuous on [a,b]

(ii) f (x) is differentiable on (a,b) and

(iii) f (a) = f (b)

Geometrical interpretation of Rolle's theorem

Let AB be the graph of y = f(x) such that A = (a_,f(a)) and B = (b_,f(b))

(i) f (x) is continuous between A and B.

(ii) f (x) has derivative between A and B i.e., there is a unqiue tangent at every point between A and B.

(iii) f (a) = f(b)

at C is parallel to x-axis.

• Langrange's Mean Value theorem: If a function f (x) is such that

(i) f (x) is continuous on [a,b] and

(ii) f (x) is differentiable on (a,b) then

Geometrical interpretation of Lagrange's Mean value theorem.

Let A(a_, f(a)) and B(b_, f(b)) be two points on the curve y = f (x) such that

(i) f (x) is continuous between A and B.

(ii) f (x) is differentiable between A and B

i.e., there is a unique tangent at every point between A and B.

then there exists a point C(c_, f(c)) between A and B such that the tangent at C is parallel to the chord AB.

Note that the point C is not unique.

Relation between dy and dy

Let A(x, y) and B(x + dx, y + dy) be two neighbouring points on the curve y = f(x).

Let dx and dy be the differentiables of x and y respectively.

AC = dx = dx

BC = dy

DC = dy

dy = f^'(x)dx

dy - dy = BC - CD = BD

\ The differential 'dy' and the increment 'dy' may not be equal.

• Generally the following points are examined for tracing the curves

(i) Symmetry about x - axis

(ii) Symmetry about y - axis

(iii) Symmetry in opposite quadrants

(iv) Symmetry about the line y = x

(v) Passage through specific points

(vi) Points of intersection with the axes

(vii) Regions where the curve is increasing or decreasing

(viii) Region of existence.