Applications of Derivatives

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1. MATHS

FORMULA

Applications of Derivatives

By

Assistant Professor (Computer Science)
Director, BST, Kokar
&

Assistant Professor (Computer Science)
“Asst, Director, BSTI, Kokar

IIT-JEE / ENGINEERING

2. Applications of Derivatives

IMPORTANT DEFINATIONS, FORMULAE AND METHODS
1. Rate of change of quantities :

C1 ut y as wih nor qx ch 2/0 ZE

represents the rate of ch represents the rate of

pot y wrt x and Y

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change of y tx at X=,

(0) If oo variables x any ner variables t, Le. F2) and

(thé by Shan Rute = a
> e
ar

Sle

Sls

Te rt change wx can elevating Das O change

y and ha of bom et
‘Hert it should be noted that z is positive ify inérenses as xincreades and is
negaive ify ocres as eres

2. Increasing Function : =
(a) Without using derivatives:

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A funetion is id vo Be rein an ter (a. Df
Hem MEH Hla) SUD fera, x/e dy
(by Using derivatives
A Sutin fis increasing on (a. D) (2) 20 foreach ia by
3 Decreasing Function ay
(a) Without using derivatives: 7
A funtion’ fi decreasing on (aD) <x, in (AS FOR) 2 FU) for

al x 2, a)

(0) Using derivative:
A function f is decreasing on (a,b) if J (3) $0 for each x in (a,b)

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3. Applications of D

4 Strictly Increas
(a) Without
‘A function f is strictly increasing on (ab) if

y, <x, in (a, D) f()< Fs) forall x € (aD)
(0) Using derivative:
A function f is strictly increasing on (a bf $ (x)>0 foreach x in (a, b
5. Stricly deereasing function
(a) Without using derivative;

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A function Ll deat (ERTEILEN

(Using derivative: >
fan fi ey decreasing on (ab) if LO for each x na).

6. Critical Point: ¢ Z

2 A point on the curve, y= f(x) where either, FO), doesn't exist or = is
calle ical pint 4

7, Method to find the intervals in which_a-function is strictly increasing or
strictly decreasing: E

1. Letihe funcion J is givén by (3) On ab.
1 Fi (x) and using YU) = 0, find all he tica points satisfying the given

interval (a,b). IF interval Is no mentioned then consider R Le (a, 4 as the

Interval. ae

x, bee

intervals

Vo The function fis sretly increasing on those intervals, in which f(x) >0 and

strictly dereasing in which f(x) <0

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4. Applications of Derivatives

2] à ll slope of tangent ote sven

8. Slope of tangent toa curve:

Let y=/(0 be a curve then m

curve at point Ph,
9. Slope of Normal toa curve:
Let y= f(x) be a curvo and Ph, k) be a point on it. Slope of tangent tothe given

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a

entre pet AAA

10. Equation tangent and normal to curve

Lei kya point he given ce =) en equation of tangent io he

(Gen gar at, is She ma h) an he equation nomial fo his curve

“ihe rs enn herve Mic PE sol anal te

(Pat any point

‘m=0, ifthe tungen is parle © x exis

It. mis not defined L = Ost the tangent is patel to years.

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IF the tangent makes an angle @ with positive x-axis, then f= tang
TV. IP tangentás parallel to a line having slope m, thes M m,
1

V. Ifthe tangent is perpendicular toa lin having slope my, them x m
VI-TWO corpo touch each other ifthe slopes of the agents are equal atthe

points of iterséction ofthe curves, =
VII. Two curves are onthogonal man, == where m, and m, be the slopes of

theic tangents atthe points of intersection ofthe curves.

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5. Jharkhand

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2 AS is cadre

Applications of Derivatives

12. Differentials:

Fora funcion y= f(x)

1. The differential of x, denoted by de, is defined by dk

I, The differential of y denoted by dy ‚is defined as dy =|

13. Approximations +
We can use the differential to approximate values of certain quantities

oo qn ds evo voy
A ee
14. Absolute Error:
Kara: O en
15, Relative Error

16, Percentage Error: >
22361006 called percentage enor in x
17, Maxima and Minima:
Let f be a function defied onan interval. The
(9 $ is ai ave a ain AN here exit a pont ein | such that
(O20), fora zer RY
‘THe number ff) is called the maximum value off E and be pin ci
called point of maximum alu of fin

(0) £ is said to have a minimum va in , I here exists a point cin I such that

SOS $00, forall rel
‘The number /(e) is called the minimum value of fin Land the point eis

called a point of minimum value of in

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6. Applications of Derivatives
18, Monotonic Function:

‘A funtion which i ether increasing or decreasing in he given interval Lis called

‘monotonic function. Every monotonic function assumes its maximunvminimom

1 ofthe function

end points ofthe domain of defi
19. Local maxima and Local minim

Let / be a function and Let e be an interior pont in is domain, then

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Ge is cal

point of local maxima if there is an h > 0 such that (6) 2 fa),
forall xe(c=h, c+

call the cal mafimun pato of /

“The number fe)
(0) is etl 3 pol of loca! minima METAN h > O such tat
MOS JO fora eh erh. x
_-The némber f(s calle the foal minimum value ff. |
20, Method t ind local maxima or Local minima using fist or second derivative
“tet: N
AM. Lettie given finite 4)
A, Find Fand sing)

ind the critical points of 40) say

ce

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Find fa) and check the sign of [CH atibese critical points. —
IV. Those points for which /7C0)>0, are known 4s points of local mininda and
orthose fa) <0 are known as points offal maxims.

V Case of Failure : I f"(2)=0 at any critical point say =X, then second
derivatives ails ni
[Now we can use first derivate test to find local maxima or minima

VI.First derivation test

(a) First check the sign of JP) ben is lit ess than x,
(b) Then secondly check the sign off (1) When x is slightly greater than x,
(©) IF JU) changes its sign fom + 10 then x, is the point of local maxima,

(2) IF JR) changes its sign from -t0 +, then x, isthe point of local minima

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7. Applications of Derivatives

point of inflexion,

(6) IF FG) does not change its sign, then x, is

means the point where there. no maxima or minima,

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Note: any mistake on this, Kindy inform on the mail

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