If y = f (x) , then dy/ dx represents instantaneous rate of change of y with respect to x .

**SUMMARY**

â€¢ If* y *=*
f *(x) , then *dy/ dx* represents
instantaneous rate of change of* y *with
respect to* x *.

If* y *=*
f *(* g *(t )) , then *dy/dt* =* f *â€² (* g *(t ))â‹…* g*â€² (t ) which is called the chain
rule.

â€¢ The
equation of tangent at ( a , b) to the curve*
y *=* f *(x) is given by* y *âˆ’*
b *= (*dy*/*dx*)_{(a,b)}(x âˆ’ a)
or *y*-b =* f *â€² (a)(*x* âˆ’ a) .

â€¢ Rolleâ€™s
Theorem

Let* f *( x) be continuous in a closed
interval [ a , b] and differentiable on the open interval ( a , b) . If* f *(*a*)
=* f *(*b*) , then there is at least one point c âˆˆ( *a,b*) where* f *â€² (c) = 0 .

â€¢ Lagrangeâ€™s
Mean Value Theorem

Let* f *( x) be continuous in a closed interval
[* a ,* b] and differentiable on the
open interval (* a ,* b) (where* f *(a) and* f *(b) are not necessarily equal). Then there is at least one point
c âˆˆ( *a,b*) such that* f *â€² (c) =* f *(b) âˆ’* f *(a) / *b-a*.

â€¢ Taylorâ€™s
series

Let* f *( *x*)
be a function infinitely differentiable at*
x *= a . Then* f *( x) can be
expanded as a series in an interval (* x *âˆ’* a , x *+ a) ,of the form

â€¢ Maclaurinâ€™s
series

In the
Taylorâ€™s series if a = 0 , then the expansion takes the form

â€¢ The
lâ€™HÃ´pitalâ€™s rule

Suppose* f *( *x*)
and* g *( *x*) are differentiable functions and* g *â€² ( *x*) â‰ 0 with

â€¢ If the
function* f *( x) is differentiable in
an open interval (* a ,* b) then we
say, if *d/dx* (* f *(x)) > 0 , âˆ€* x *âˆˆ( *a,b*) then* f *( x)* *is strictly increasing in the interval (*a,b*).

â€¢ if *d/dx* ( *f*
(*x*)) <
0 , âˆ€ *x*
âˆˆ(
*a*,*b*)
then *f* ( *x*) is strictly decreasing in the
interval ( *a* , *b*)

â€¢ A
procedure for finding the absolute extrema of a continous function *f* ( *x*)
on a closed interval [*a *,* b*].

Step 1 :
Find the critical numbers of *f* (
*x*) in ( *a* , *b*)
.

Step 2 :
Evaluate *f* (
*x*) at all critical numbers and at the
endpoints *a* and *b* .

Step 3 :
The largest and the smallest of the values in Step 2 is the absolute maximum
and absolute minimum of *f* (
*x*) respectively on the closed interval
[*a* , *b*]
.

â€¢ First
Derivative Test

Let (
*c* , *f* (*c*)) be a critical point
of function *f* (
*x*) that is continuous on an open
interval *I* containing *c* . If *f* ( *x*)
is differentiable on the interval, except possibly at *c* , then *f* (*c*) can be classified as
follows:(when moving across *I* from
left to right)

(i) If *f* â€² ( *x*)
changes from negative to positive at *c*
, then *f* (
*x*) has a local minimum *f*(*c*).

(ii) If *f* â€² ( *x*)
changes from positive to negative at *c*
, then *f* (
*x*) has a local maximum *f *(*c*).

(iii) If
*f* â€² ( *x*)
is positive on both sides of *c* , or
negative on both sides of *c* then *f* ( *x*)
has neither a local minimum nor a local minimum.

â€¢ Second
Derivative Test

â€¢ Suppose
that *c* is a critical point at which *f* â€² (*c*)
=
0 , that *f* â€²â€²
(
*x*) exists in a neighbourhood of *c *, and that* f *â€²* *(*c*)* *exists.
Then* f *has a relative maximum value
at* c *if* f *â€²â€²* *(*c*)* *<* *0 and a* *relative minimum value at *c*
if *f* â€²â€² (*c*) >
0 . If *f* â€²â€²
(*c*) =
0 , the test is not informative.

Tags : Applications of Differential Calculus | Mathematics , 12th Maths : UNIT 7 : Applications of Differential Calculus

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