Click to Chat

1800-1023-196

+91-120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: Rs. 15,900
• View Details

Revision Notes on Application of Derivatives

The concepts of straight line, maxima and minima, global maxima and minima, Rolle’s Theorem and LMVT all come under the head of Application of Derivatives.

If a function is increasing on some interval then the slope of the tangent is positive at every point of that interval due to which its derivative is positive.

Similarly, the derivative of a function which is decreasing on some interval is negative as the slope of the tangent is negative at every point of that interval.

A function f is said to have a local maximum (also termed as relative maximum) at x = a if f(x) ≤ f(c), for every x in some open interval around x = c.

A function f is said to have a relative minimum or a local minimum around x = c if f(x) ≥ f(c), for every x in some open interval around x = a.

A function f is said to have a global maximum (also termed as absolute maximum) at x = a if f(x) ≤ f(c), for every x in the domain under consideration.

A function f is said to have an absolute minimum or a global minimum around x = c if f(x) ≥ f(c), for every x in the whole domain under consideration.

Rolle’s Theorem

Let y = f (x) be a given function which satisfies the conditions:

1) f (x) is continuous in [a , b]

2) f (x) is differentiable in (a , b)

3) f (a) = f (b)

Then f'(x) = 0 at least once for some x ∈ (a, b).

Certain points to be noted in Rolle’sTheorem include:

Converse of the theorem does not hold good.

There can be more than one such c.

The conditions of Rolle’s Theorem are only sufficient and not necessary.

Lagrange Mean Value Theorem (LMVT)

If a given function y = f (x) satisfies certain conditions like:

f(x) is continuous in [a , b]

f(x) is differential in (a, b)

then f'(x) = [f(b) – f(a)]/[b–a] for some x ∈ (a, b). This is the generalization of the Rolle’s Theorem and is termed as Lagrange Mean Value theorem.

A function is said to be monotonically increasing at x = a if f(x) satisfies f(a+h) > f(a) and f(a-h) < f(a), for some small positive h.

A function is said to be monotonically decreasing at x = a if f(x) satisfies f(a+h) < f(a) and f(a-h) > f(a), for some small positive h.

If f'(x) > 0 ∀ x ∈ (a,b) and points which make equal to zero (in between (a, b)) don’t form an interval, then f (x) would be increasing in [a, b] otherwise it will be non-decreasing function.

If f'(x) > 0 ∀ x ∈ (a,b) and points which make equal to zero (in between (a, b)) don’t form an interval, f (x) would be decreasing in [a, b], otherwise it will be non-increasing.

For all x and y, such that x≤ y, if f(x) ≤ f(y), then the function f is said to be monotonically increasing, increasing or non-decreasing.

Similarly, for x ≤ y, if f(x) ≥ f(y), then the function is monotonically decreasing, decreasing or non-increasing i.e. it reverses the order.

If f is increasing for x > a and f is also increasing for x < a then f is also increasing at x = a provided f(x) is continuous at x = a.

If f(x) is strictly increasing, then f-1 exists and is also strictly increasing.

If f(x) is strictly increasing on [a, b] and is also continuous then f-1 is continuous on [f(a), f(b)].

If f(x) and g(x) are strictly increasing (decreasing) functions on [a, b], then gof(x) is strictly increasing (decreasing) function on [a, b].

If one of the two functions f(x) and g(x) is strictly increasing and other is strictly decreasing then gof(x) is strictly decreasing on [a, b].

If a continuous function y = f(x) is strictly increasing in the closed interval [a, b], then f(a) is the least value.

If f(x) is decreasing in [a, b], then f(b) is the least and f(a) is the greatest value of f(x) in [a, b].

If f(x) is non-monotonic in [a, b] and is continuous then the greatest and the least value of f(x) in [a, b] are those where f(x) = 0 or f’(x) does not exist or at the extreme values.

The direction of acceleration is in the direction of velocity or opposite to it.

When the particle is going upward, the value of g is negative and when it is coming back, the value of g is positive.

At maximum height the velocity of a particle is zero. The value of g is 9.8 m/s2 or 980 cm/s2.

Slope of tangent to the curve y = f(x) at the point (x, y) is m = tan θ = [dy/dx](x,y)

If the equation of the curve is in the parametric form x = f(t) and y = g(t), then the equations of the tangent and the normal are y – g(t) = g'(t)/f'(t)(x–f(t)) and f'(t)[x–f(t)] + g'(t) [y–g(t)] = 0 respectively.

The equation of tangent to the curve y = f(x) at the point P(x1, y1) is given by y – y1 = [dy/dx](x,y) (x – x1)

If dy/dx = 0 then the tangent to curve y = f(x) at the point (x, y) is parallel to the x-axis.

If dy/dx → ∞, dx/dy = 0, then the tangent to the curve y = f(x) at the point (x, y) is parallel to the y-axis.

If dy/dx = tan θ > 0, then the tangent to the curve y = f(x) at the point (x, y) makes an acute angle with positive x-axis and vice versa.

If two curves are orthogonal, then the product of their slopes is -1 everywhere wherever they intersect.

Length of tangent, normal, subtangent, subnormal:

Tangent = 

Subtangent = 

Normal = 

Subnormal = 


### Course Features

• 728 Video Lectures
• Revision Notes
• Previous Year Papers
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Test paper with Video Solution