PDF Introduction to Calculus - Some useful formulas - University of Waikato

Introduction to Calculus

– Some useful formulas

Quadratics

If ax²+bx+c=0 then x= !b± b²!4ac 2a Binomial Theorem

(a+b)ⁿ = n 0

!

"#

$

%&aⁿ+ n

1

!

"#

$

%&a^n'1b¹+ n

2

!

"#

$

%&a^n'2b²+…+ n

r

!

"#

$

%&a^n'rb^r +…+ n

n

!

"#

$

%&bⁿ

n r

!

"#

$

%& =nC_r = n!

n'r

( )

^!r!

Binomial Expansions Taylor Expansions a+b

( )

^{2 =a2+2ab+b2 for n!0 : f(a+h)= f(a)+ f(j)(a)}

j=1 j!

"

n ^{hj + f}_(n⁽ⁿ⁺¹⁾_+1)!

^{( )}

^{# hn+1}

a+b

( )

^{3=a3+3a2b+3ab2+b3}

a+b

( )

^{4 =a4 +4a3b+6a2b2+4ab3+b4}

Identities Products

cos²!+sin²! =1 2 sinAcosB=sin(A+B)+sin(A!B)

tan²!+1=sec²! 2 cosAsinB=sin(A+B)!sin(A!B)

cot²!+1=cos ec²! 2 cosAcosB=cos(A+B)+cos(A!B)

2 sinAsinB=cos(A!B)!cos(A+B)

Compound Angles Sums

sin

(

A+B

)

^{=sinAcosB+cosAsinB sinC+sinD=2 sinC+D}

2 cosC!D 2 cos

(

A+B

)

^{=cosAcosB!sinAsinB sinC!sinD=2 cosC+D}

2 sinC!D 2 tan

(

A+B

)

^{= tanA+tanB}

1!tanAtanB cosC+cosD=2 cosC+D

2 cosC!D 2 cosC!cosC=2 sinC+D

2 sinC!D 2

OVER ....

Double Angles General Solutions

sin 2A=2 sinAcosA If sin! =sin" then ! =n#+($1)ⁿ",n%Z tan 2A= 2 tanA

1!tan²A If cos! =cos" then ! =2n#±",n$Z

cos 2A=cos²A!sin²A If tan! =tan" then ! =n# +",n$Z =2cos²A!1

=1 - 2 sin²A Differentiation

y= f(x)

dy

dx = f!(x)

c 0

x^{n nxn!1, n"R}

lnx ¹x

e^ax ae^ax

a^x a^xln a

sinx cosx

cosx !sinx

tanx sec²x

secx secxtanx

co secx !co secxcotx

cotx !co sec²x

sin^!1

( )

_a^{x a21}!x²

tan^!1

( )

_a^x a²+^ax²

ln cosx !tan(x)

Product Rule Quotient Rule

( f !g)"= "f !g+ f ! "g f g

!

"#

$

%&

' = g( 'f ) f ( 'g g²

Composite Function Rule Logarithmic Differentiation

f (g(x))!= !f (g(x)).g (x)!

(

ln f(x)z

)

^{! = f}_f^!_(x)^(x)

or if y= f(u) and u=g(x) then dy dx = dy

du!du dx