diff cal rules eng

8/10/2019 Diff Cal rules eng

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TOPIC: DIFFERENTIAL CALCULUS

Algebraic Functions

(c) = 0

(cu) = c

= u + v

=

(un) = n un-1

Logarithmic & Exponential Functions

(ln u) =

(loga u) =

(eu) = eu

(au) = aulna

Trigonometric Functions

(sin u) = cos u

(cos u) = -sin u

(tan u) = sec2u

(cot u) = -csc2u

(sec u) = sec u tan u

(csc u) = -csc u cot u

Hyperbolic Functions

(sinh u) = cosh u

(cosh u) = sinh u

(tanh u) = sech2u

(coth u) = -csch2u

(sech u) = -sech u tanh u

(csch u) = -csch u coth u

Slopes of the Curve( y f(x! !

Slope at any point of = y =

"ate of Change

The derivative of the function is

identical to its rate of change

= V = velocity

= = a = acceleration

Curvature an# "a#ius of Curvature

=

! = 1" =

$raph of a function y f(x!

if y = 0 and y# is negative$ the point

is a %a&i%u% point ( concave

do'n'ard)

if y = 0 and y# is positive$ the point

is a %ini%u% point ( concave

up'ard)

a point of inflection is a point at'hich the curve changes fro%

concave up'ard to concave

do'n'ard and vise-versa

at point of inflection y# = 0

Steps in Solving %axima %inima roblems1 dentify the varia*le to *e %a&i%ie

or %ini%ie$ say area ,


2/4

2 &press this varia*le in ter%s of

other relevant varia*le(s)$say , =

f(&$y). f the function shall consist of %ore

than one varia*le$ e&press it in ter%s

of one varia*le (if possi*le) using theconditions in the pro*le%$ say , =

f(&)

/ ifferentiate and euate to ero$d,"d& = 0

S,34 3!564S7

1 valuate7

a2 * 1"8c / d 8

2 valuate7

a2 * 1"8

c / d 8

. 9hat is the slope of the curve &2+ y2: ;& : 10y+ 8 = 0 at (1$0)

a2"8 * 8"2

c -2"8 d -8"2

/ ; $ 11"2/ )

c (1"8$ 1"?) d (1".$ 11"2@)

@


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a10 fps * 12 fpsc 1. fps d 18 fps

1 The height of a circular cylinder is 80 in anddecreases at the rate of / in"sec$ 'hile the radiusat the *ase is 20 in and increases at a rate of 1

in"sec ,t 'hat rate is the volu%e changing

a28@1 * @812

c 8@21 d 128@20 9ater is pouring into a s'i%%ing pool ,fter t

hours there are t + t1"2gallons in the pool ,t 'hatrate is the 'ater pouring into the pool 'hen t =

hours

a@"; gph * ;"@ gphc 8"; gph d ;"8 gph

21 T'o post 10% and 18% high stands .0% apartThey are to *e stayed *y a trans%ission 'iresattached to a single stae at the level ground$ the

'ires running to the top of the posts 9hereshould the stae *e place fro% the 10% pole to

used the least a%ount of 'ire

a * 10c 11 d 12

22 9ater is pouring into a conical cistern at a rate of? %."%in f the height of the inverted cone is12% and radius of its circular opening is ;%

Ao' fast is the 'ater level rising 'hen the 'ater

is /% deepCa 0/; * 08;

c 0;/ d0.;

2. , cylindrical glass Dar has a plastic top$ if theplastic is half as e&pensive as glass per unit area$

find the %ost econo%ical proportions for the Darah = ."2 r * h = 2". r

c h = 1". r d h = > r

2/ , rectangular field of given area is to *e fenced

off along the *an of a river f no fence isneeded along the river$ 'hat is the shape of the

rectangle reuiring the least a%ount of fencinga & = 8y * & = /y

c & = .y d & = 2y

28 Victory liner agrees to transport 100 people orless fro% station , to station 6 for 318 each

passenger t 'as furthered agreed that is there

are %ore than 100 passengers$ Victory liner 'illreduce the fair for every passenger *y 3010 in

e&cess of 100 passenger 9hat nu%*er ofpassengers 'ill produce the %a&i%u% revenue

for the *us co%pany

a 100 * 118c 120 d 128

S'LE%ETA") "*+LE%S,

1


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18 , poster is to contain .00 %2 of printed %atter 'ith%argins of 10 c% at the top and 8 c% at each side

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