Differential Calculus To err is to admit to forgive to blame it on others to repeat ANONYMUS human, superhuman, divine, politics, unprofessional. Calculus is a central branch of Mathematics, developed from algebra and geometry. It is built on two major complementary ideas, both of which rely critically on the concept of limits.
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Differential Calculus
To err is to admit to forgive
to blame it on others to repeat
ANONYMUS
human, superhuman, divine,
politics, unprofessional.
Calculus is a central branch of Mathematics, developed
from algebra and geometry.
It is built on two major complementary ideas, both of
which rely critically on the concept of limits.
The first is the differential calculus ( Part A ) , which is
concerned with the instantaneous rate of change of quantities
with respect to other quantities.
More precisely, the local behavior of functions which can
be illustrated by the slope of a function's graph.
The second is the integral calculus ( Part B ) , which studies
the accumulation of quantities, such as areas under a curve,
linear distance traveled, or volume displaced.
These two processes act inversely to each other, a fact
delivered conclusively by the Fundamental theorem of
calculus.
nth order derivatives of some standard functions:
1. y = eax
2ax 2 ax
2 2y = a.a e = a e
d y
dx=
ax
1y = a e dy
dx=
yn = an eax.
.
.
2. y = amx where m is a positive integer.
y1 = D (amx) = m amx log a.
y2 = m log a . (m amx log a)
= (m log a)2 amx
yn = (m log a)n amx.
.
.
y3 = (m log a)3 amx
3. y = (ax + b)m, where m is a positive integer such that m > n .
y1 = m(ax + b)m-1.a
y2 = m(m – 1)(ax + b) m-2. a2
y3 = m(m – 1) (m- 2) (ax + b)m-3 a3
yn = m(m- 1) (m – 2) … [m – (n – 1)] (ax + b)m-n an.