8/10/2019 Lecture 1 Basic Calculus

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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**'

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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

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