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Title: Application of derivatives
Description: Easy to understand notes of Application of derivates.
Description: Easy to understand notes of Application of derivates.
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Mathematics Notes for Class 12 chapter 6
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The equation of normal at (x, y) to the curve is
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Slope of Tangent
(i) If the tangent at P is perpendicular to x-axis or parallel to y-axis,
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(ii) If the tangent at P is perpendicular to y-axis or parallel to x-axis,
Slope of Normal
(ii) If
, then normal at (x, y) is parallel to y-axis and perpendicular to x-axis
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Length of Tangent and Normal
(i) Length of tangent, PA = y cosec θ =
(ii) Length of normal,
(iii) Length of subtangent,
(iv) Length of subnormal,
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Angle of Intersection of Two Curves
Let y = f1(x) and y = f2(x) be the two curves, meeting at some point P (x1, y1), then the angle
between the two curves at P (x1, y1) = The angle between the tangents to the curves at P (x1, y1)
The other angle between the tangents is (180 — θ)
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∴ The angle of intersection of two curves θ is given by
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Derivatives as the Rate of Change
If a variable quantity y is some function of time t i
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, y = f(t), then small change in Δt time At
have a corresponding change Δy in y
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So, the differential coefficient of y with respect to x i
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, (dy/dx) is nothing but the rate of
increase of y relative to x
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f is continuous in the closed interval [a, b]
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ncerthelp
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f(x) is differentiable in the open interval (a, b)
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f(a)= f(b)
Then, there is some point c in the open interval (a, b), such that f‘ (c) = 0
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The conclusion is that there is at least one point c between a and b, such that the tangent to the
graph at (c, f(c)) is parallel to the x-axis
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Lagrange’s Mean Value Theorem
Let f be a real function, continuous on the closed interval [a, b] and differentiable in the open
interval (a, b)
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Remarks In the particular case, where f(a) = f(b)
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Thus, when
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f(a) = f (b), f ‗ (c) = 0 for some c in (a, b)
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Approximations and Errors
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Let Ax denotes a small increment in Δx, corresponding
which y increases by Δy
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Let Δx be the error in the measurement of independent variable x and Δy is corresponding
error in the measurement of dependent variable y
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Monotonic Function
A function f(x) is said to be monotonic on an interval (a, b), if it is either increasing or
decreasing on (a, b)
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Strictly Increasing Function
f(x) is said to be increasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) > f(x2)
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ncerthelp
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Non-Decreasing Function
f(x) is said to be non-decreasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) ≥ f(x2)
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It means that
there is a certain decrease in the value c f(x) with an increase in the value of x
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ncerthelp
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Non-increasing Function
f(x) is said to be non-increasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) ≤ f(x2)
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If a function is either strictly increasing or strictly decreasing, then it is also a monotonic
function
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(ii) A function f (x) is said to be increasing on [a , b], if it is increasing (decreasing) on (a ,b)
and it is also increasing at x = a and x = b
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ncerthelp
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∴
(iv) Let f be a differentiable real function defined on an open interval (a, b)
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If f ‗ (x) < 0 for all x ∈ (a , b), then f (x) is decreasing on (a, b)
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If f ‗(x) > 0 for all x ∈ (a, b) except for a finite number of points, where f ‗ (x) = 0, then
f(x) is increasing on (a, b)
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Properties of Monotonic Functions
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Maxima and Minima of Functions
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If f(x) ≤ f(a) for all x ∈
(a – h, a + h), where h is somewhat small but positive quantity
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ncerthelp
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2
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The point x = a is called a point of minimum of the function f(x) and f(a) is known as the
minimum value or the least value or the absolute minimum value of f(x)
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If f(x) is continuous function in its domain, then at least one maxima and one minima
must lie between two equal values of x
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Maxima and minima occur alternately, i
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, between two maxima there is one minima
and vice-versa
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If f(x) → ∞ as x → a or b and f ‗ (x) = 0 only for one value of x (sayc) between a and b,
then f(c) is necessarily the minimum and the least value
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If f(x) → p -∞ as x → a or b and f(c) is necessarily the maximum and the greatest value
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If f(x) be a differentiable functions, then f ‗(x) vanishes at every local maximum and at
every local minimum
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The converse of above is not true, i
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, every point at which f‘ (x) vanishes need not be a
local maximum or minimum
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g
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The function
has neither minimum nor maximum
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11 | P a g e
3
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g
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Thus, the maximum and minimum values of f(x) defined above are not necessarily the
greatest and the least values of f(x) in a given interval
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A minimum value at some point may even be greater than a maximum values at some
other point
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By a local maximum (or local minimum) value of a
function at a point c ∈ [a, b] we mean the greatest (or the least) value in the immediate
neighbourhood of x = c
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A function may have a number of local
maxima or local minima in a given interval and even a local minimum may be greater than a
relative maximum
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Local Minimum
f (x) > f(a), ∀ x ∈ (a – δ, α + δ), x ≠ a
or f(x) – f(a) > 0, ∀ x ∈ (a – δ, α + δ), x ≠ a
In such a case f(a) is called the local minimum value of f(x) at x = a
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First Derivative Test
Let f(x) be a differentiable function on an interval I and a ∈ I
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(i) Point a is a local maximum of f(x), if
(a) f ‗(a) = 0
(b) f ‗(x) > 0, if x ∈ (a – h, a) and f‘ (x) < 0, if x ∈ (a, a + h), where h is a small but
positive quantity
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ncerthelp
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(ii) Point a is a local minimum of f(x), if
(a) f ‗(a) = 0
(b) f ‗(a) < 0, if x ∈ (a – h, a) and f ‗(x) > 0, if x ∈ (a, a + h), where h is a small but
positive quantity
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(iii) If f ‗(a) = 0 but f ‗(x) does not changes sign in (a – h, a + h), for any positive
quantity h, then x = a is neither a point of minimum nor a point of maximum
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Second Derivative Test
Let f(x) be a differentiable function on an interval I
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Then,
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And if f iv > 0,
then it is a local minimum
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If n is even and f n (a) > 0 ⇒ x = a is a point of local minimum
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Important Points to be Remembered
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Then, range of
value of f(x) for x ∈ [a, b] is [m, M]
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To Check for the injectivity of a Function A strictly monotonic function is always oneone (injective)
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Thus, a function attains an
extreme value at x = a, if f(a) is either a local maximum value or a local minimum value
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13 | P a g e
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It
is not sufficient
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e
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There are functions for which the derivatives vanish at a point but do not have an
extreme value
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g
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The values of x for which f
‗(x) = 0 are called stationary values or critical values of x and the corresponding values
of f(x) are called stationary or turning values of f(x)
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Maximum and minimum values of a function f(x) can occur only at critical points
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Thus, the points where maximum or minimum value occurs are necessarily critical
Points but a function may or may not have maximum or minimum value at a critical point
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At x = 0, f ‗(x)= 0
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Such point is called
point of inflection, where 2nd derivative is zero
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Now, f ―(x)= 0 when x = nπ, then this points are called point of inflection
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It is not necessary that 1st derivative is zero
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2nd derivative must be zero or 2nd derivative changes sign in the neighbourhood of point
of inflection
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Let [a, b] ⊆ D, then global maximum/minimum of f(x) in [a, b] is basically the
greatest/least value of f(x) in [a, b]
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Global Maximum/Minimum in [a, b]
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14 | P a g e
In order to find the global maximum and minimum of f(x) in [a, b], find out all critical points
of f(x) in [a, b] (i
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, all points at which f ‗(x)= 0) and let f(c1), f(c2) ,…, f(n) be the values of the
function at these points
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and M1 → Global minima or least value
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Title: Application of derivatives
Description: Easy to understand notes of Application of derivates.
Description: Easy to understand notes of Application of derivates.