Differentiation and Trigonometric Functions, Exams of Biology

Download Differentiation and Trigonometric Functions and more Exams Biology in PDF only on Docsity!

N◦in

sing

=\

tim ¥×

Is ✗→^0 ✗to 1% = 0 Note switch to^ sin^ and^ cos^!^! COSSX (^) • 5 ✗to 1 Exampled a

! " " cos ◦^ •^ s^ =^ 5 i )¥%%¥ = 1in^ {i;÷

. 5 =^ 1.5-- . ✗ (^) → 0

in tin^ ᵗ;÷→ [ T.IE#-i-ii.i--Bo-o--oiIoI?!:--f=lim(sino+ocoso-g

• →^ ◦ sin÷ = 1in "ⁿ%:s¥+o
• →^0 = *%^ %i%%• . Eso = 1in^ " . ÷.

'

÷

On- nt÷ˢ¥¥%"¥÷É¥ =g sihcsinx) sin

.^ =^ 1.^ = (^1).

u is 1in^ s¥×É✗

✗-I 0

= l^ im ✗→ (^) ◦ Ex^

• ¥ - ÉiÉ¥¥ % ¥×%÷×^

¥

.^ ×

II.◦ ¥. 5 ¥ . ÷ ✗ = E (^) E (^) % ! f.t.t.

Examples find^ y ' : ily = ant ( It^ tant) . csinti-sint.CI/-tantIddYt--- (I^ +^ tant) " = (It tant) cost^ -^ Sint seit I (^) ( It^ tant^ ) '

cost +

ˢY¥

. (^) cost -^ Sint.

( I^ +^ tant) '

cost +^ Sint^

• %¥ , ii.y.II.ci?:---:-=:scx+i*i--3icscx-x3C-cscxcotx)--3X~cscx- ✗ 3 cscxcotx

iii) y = ✗ " é see ✗ r 4 ¥ (^) :(

✗Ye

✗ sea (^) + (^) ✗ ecx tie✗ = 2 ✗ (^) e ✗ see (^) ✗ + ✗ ' e ✗ secxtxexsecxtanx ✗e✗secx(2t✗+✗ta

v ¥n( ✗E)^ , y

= ✗ ex

so

y = (^) ✗ ex^ y=y ' y

ex + ✗^ e ✗ = (^ it^ ×^ ) e^ ' y

= e^ ✗

• (^ I^ -1×^1 e✗=e✗+e✗+✗e✗ = (^) @ + He

,÷I÷÷÷÷::* = 3e✗+xe✗

=/ 3 +^ ×,

ex j"=cn- : (^) % ,

v4 (^) flt / (^) = et ' Yt ᵈ¥=eÉ '""

=eᵗ%ˢ"ᵗ[ zteosyt-te-sihyt.it vii) 91 × 1 = sin (^ a^ >

1. , %, = costa^ > " 1.67× 9 = cos^ (^ at)^ a. 7 ×(3×2) = 3A^ In^ ? I > × > cos ( a > × )

Examples find^ d¥ it ✗ (^) Y >

• (^) e (^7) = cos (^) ( (^) sing ) ( " °) so

ftp.dnet

rule 2.✗ y 't (^) ×?^ 351 ¥

e ' dy _

• singing)^. Cosy d

3 × 54 × 1

e' off

• singing^4 ¥ = -2× 5 -2 (^) ✗ (^) y ] ) ᵈd¥ = z×~y~+eY + sinking) cosy

Equation (^) of tangent^ line at (^ 1,0 )^ : slope : m=&%|u , o^ ) = ;¥¥+sinisi = 9- =

y

-^ 0=0^ (^ x^ - 1) y E9tkIi: ( (^) 1,

tangent

✗ =^1

9= ✗-

Examplei-Diferntiate.it y= In (^) ( 15 sink) Inca b) =/natlnb

In

(F)

=/na - lnb

= 1h15^ - Infinity 1n(a7=n, = (^) In (^) 15th ( (^) simp = his -12 (^) Infinity , d¥=

1. t.com^ ( thx )=& sihx lh⑨M=g¥ 9 ¥ _2-

function

city :(sinx^ ) " ( function ↓ (^) In ˢny=Ñi×ii E- logarithmic differentiate lny-ri-%a1.tr.ie ↓ see (^) the tyd-a-zxhsinx-i-s.fm

" " notes

ᵈ¥=y[ zxlnsinx -1159¥;] =%in×ji[ zxlnxtxcotx (^) ]

iii) y = Nx [iGrithmiz ↑ S differentiate lay: In six Iny= ux'+SexY^ -Intanx my =3lux +^ * InCxY

-^ In^ tan ↓ &o

tx. sex if i tir-FIT ityE ·

jux.is

sec (^) sx (^) <- (^) Etsinx

S ↓

f

s

• 1n3.34Tex)^ +^ 3x " Inw=suttanxx => Inw=bux.x wow = (^) * Intanx (^) + lux^ to six dw= Htaux^ Hose The = (^) Hanxs* [Heux

sein