8/10/2019 Lecture 1 Basic Calculus
1/3
FIITJEE Limited
Lecture Plan
Lecture 1 Basic Calculus
Concepts:Position, Displacement, Distance, Average Velocity, Average Speed, Instantaneous
Velocity, Meaning of instantaneous, Derivative as Limit,Standard Differentiation Formulae, Differentiation Rules, Multiplied constant, Sum
rule, Product rule, Quotient rule, !ain rule, Derivative as Slope of a urve, Maxima
and Minima, Integration as reverse process of differentiation "Anti derivative#,
Standard Integration Formulae, Integration of f"a$%' Definite Integration,
Integration as Sum of small parts &et(een t!e integration limits, Integration as Area
under t!e curve, Definite Integration'
Differentiation Integration Differentiation Integration
)n nd x nxdx
=)
)
nn xx dx C
n
+
= +
+
csc csc cotd
x x xdx
= csc cot cscx xdx x C= +
( ) *d
Cdx
=+Cdx Cx C = + )lnd x
dx x=
)lndx x C
x= +
( )sin cosd
x xdx
= cos sinx dx x C= + cot cscd x xdx
= csc cotxdx x C= +
cos sind
x xdx
= sin cosxdx x C= + tan secd x xdx
=sec tanxdx x C= +
sec sec tand
x x x
dx
= sec tan secx xdx x C= + x xd e edx
= x xe dx e=
Product Rule: Quotient Rule:
( ) ( )( ) ( ) ( ) ( ) ( )) ) )' ' 'd d d
f x f x f x f x f x f xdx dx dx
= + ( )
( )
( ) ( ) ( )
( ){ }
) ) )
' " #'d d
f x f x f x f xf xd dx dx
dx f x f x
=
( ) ( )(!ered d
f ax b a f X X ax bdx dX
+ = = + ( ) ( ))
(!eref ax b dx f X dX X ax ba
+ = = +
Problems:
Q1. )' Find t!e derivative of t- at t . )*'
' Find t!e derivative of //t at t . l**'
8/10/2019 Lecture 1 Basic Calculus
2/3
0' Find t!e derivative of t at t . )'
1' For some constants a and &, find t!e derivative of
"i# "t 2a# "t2 "ii# "at% "iii# "t2a#3"t2
4' Find t!e derivative of "tn2an#3"t2a#
for some constant a'
5' Find t!e derivative of
"ii# "4t
0
%0t
2)# "t2)#"iii# "t20# "4t0%0t# "iv# t4"0t25t2/#
"v# "t21#"0t21t24# "vi#
) 0 )
t
t t
+
6' Find t!e derivative of t!e follo(ing functions7
"i# sin t cos t "ii# sec t "iii# 4sec t % 1cos t
"v# 0cot t % 4cosec t
"vi# 4sint25cost%6 "vii# tan t 2 6sec t
Q2. Differentiation &y c!ain rule7
"i# sin "$% 4# "ii#' cos "sin $# 0"iii# sin "a$ %
"iv# sec "tan " $ ## "v#'
( )
( )
sin
cos
at b
ct d
++
"vi# cos"$0#sin"$4#
"vii#
( ) cot t"viii#
( )cos t
Q. "Derivative as rate of !ange#
Find t!e rate of c!ange of t!e area of a circle per second (it! respect to its radius r
(!en r . 4 cm and it is increasing at a rate of )cm3s'
Ans7)*8 cm3s
Q!. 9!e volume of a cu&e is increasing at a rate of / cu&ic centimetres per second' :o(
fast is t!e surface area increasing (!en t!e lengt! of an edge is )* centimetres;
Ans70'5cm3s
Q". A stone is dropped into a
8/10/2019 Lecture 1 Basic Calculus
3/3
Top Related