Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 8 Application of Derivatives will enable students to study smartly.
CHSE Odisha 12th Class Math Notes Chapter 8 Application of Derivatives
Tangents and normals:
(a) If y = f(x) is the equation of any curve then = The slope of the tangent at P(x1, y1).
(b) Slope of the normal at (x1, y1)
(c) Equation of tangent at P(x1, y1) is y – y1 =
(d) Equation of normal at P(x1, y1) is y – y1 =
(e) Angle between two curves is the angle between two tangents at the point of contact.
Increasing and decreasing functions:
If y = f(x) is defined in [a, b] then
(i) f'(x) > 0, x ∈ (a, b)
⇒ f is strictly increasing on (a, b).
(ii) f'(x) > 0, x ∈ (a, b)
⇒ f is monotonic increasing on (a, b).
(iii) f'(x) < 0, x ∈ (a, b)
⇒ f is strictly decreasing on (a, b).
(iv) f'(x) < 0, x ∈ (a, b)
⇒ f is monotonic decreasing on (a, b).
(v) f'(x) = 0, x ∈ (a, b)
⇒ f is a constant function on (a, b).
Approximation:
(a) If y = f(x) is a function and δx is a very small change in x then the respective change in y is δy given by
δy = f'(x) δx => dy = f'(x) δx.
∴ The approximate value of y = f(x) at
x = a + δx is f(a + δx)
= f(a) + f(a) δx
Maxima and minima:
(a) First derivative criteria to find max/min of y = f(x)
Algorithm:
Step-1 : Put and solve for x.
Let x = a, b, c ……
Step-2 : If changes sign from (+ve) to (-ve) in then at x = a, ‘f’ has a local maximum. If changes sign from (-ve) to (+ve) in then at x = a, f has a local minimum. If in thenat x = a ‘f’ has neither maxima nor a minima (it may be a point of inflexion).
(b) Second derivative criteria
Algorithm:
Step-1 : Find the roots of f'(x) = 0.
Let they are a, b, c ……
Step-2 : Find f”(x) and put x = a, b, c ……
(i) If f”(a) > 0 then at x = a, f has a local minimum.
(ii) If f”(a) < 0 then at x = a, f has a local maximum.
(iii) If f”(a) = 0 and f”(x) changes sign in (a – δ, a + δ) then x = a is a point of inflexion.
(iv) If f”(a) = 0 and f”(x) does not change sign in then use first derivative criteria to check for maxima/minima.
Mean Value Theorems:
(a) Rolle’s theorem:
If a function f is
(i) continuous on the closed interval [a, b]
(ii) differentiable on the open interval (a, b) and
(iii) f(a) = f(b) then there exists a point c ∈ (a, b) such that f(c) = 0.
Geometrical interpretation:
If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b) then there exists atleast one point c ∈ (a, b) such that at x = c the tangent is parallel to x-axis.
Algebraic interpretation:
Between two roots ‘a’ and ‘b’ of f(x) there exists atleast one root of f'(x).
(b) Cauchy’s Mean Value theorem:
If ‘f’ and ‘g’ are two functions such that
(i) both are continuous on [a, b]
(ii) both are differentiable on (a, b) and
(iii) g'(x) ≠ 0 for any x ∈ (a, b) then there exists atleast one point c ∈ (a, b) such that
Geometrical interpretation:
The conclusion of Cauchy’s theorem can be written as i.e. the ratio of the mean rate of increase of two functions in an interval equals to the ratio of actual rate of increase at some point of the interval.
(c) Lagrange’s Mean Value theorem:
If a function f is
(i) continuous on the closed interval [a, b]
(ii) differentiable on the open interval(a, b) then there exists atleast one c ∈ (a, b), such that.
Geometrical interpretation:
Between two points A and B of the graph of y = f(x) there exists atleast one point c such that the tangent is parallel to the chord AB.
Indeterminate forms & L’Hospitals rule:
A limit is said to be in indeterminate form if it takes any of the forms.
Note:
If a limit is in indeterminate form then it can be evaluated using the following methods.
(i) Change the function to determinate form (by rationalisation, expansion or any other means) then find the limit.
Or, (ii) Bring to form then use L’Hospitals rule.
L’Hospitals rule:
Let f and g are two functions differentiable on some open interval containing ‘a’ such that g'(x) ≠ 0 for x ≠ a and g(a) = f(a) = 0, then provided the latter limit exists.