FM Methods in calculus name _______________________

Objective Deadlines / Progress

Met

hods

in c

alcu

lus e

xpan

sion

evaluate improper integrals; find the area under a graph where a bound is infinity or undefined

Know how to set out working using limits Notation

Where both bounds are undefined / or a value in between is undefined – find area by splitting into two integrals and using limits

Mea

n of

a fu

nctio

n

Find the mean of a function

using f= 1b−a∫a

b

f (x )dx

Understand how the mean is affected when the function is transformed by for example f ( x )+A or −f ( x ) or B f ( x )

Inve

rse

Trig

onom

etric

fun

ction

s

Integrate inverse trig functions using a substitution method and trig identities;

Know the standard integrals for

∫ 1√a2−x2

dx and ∫ 1a2+x2

dx

Integrate using partial fractions

Differentiate inverse trig functions

FM Methods in calculus name _______________________

Improper integrals

Notes

A definite integral represents the area enclosed by a continuous function y=f (x ), the x-axis and line x=a and x=b

An Improper integral represents the area where one of the limits is infinite or where the function is not defined at some point

The integral ∫a

b

f (x )dx is improper if

1) one or both of the limits is infinite2) f(x) is undefined at x=a or x=bor

at a point in the interval [a, b]

FM Methods in calculus name _______________________

WB A1 limits of inifinity Evaluate each improper integral

a¿∫1

∞ 1x2dxb¿∫

1

∞ 1xdx

WB A2 undefined limits Evaluate each improper integral

a¿∫1

0 1x2dxb¿∫

0

2 x√4−x2

dx

FM Methods in calculus name _______________________

WB A3 both limits are infinity a) Find ∫ x e−x2

dx

b) Hence show that ∫−∞

∞

x e−x2

dx converges and find its value

FM Methods in calculus name _______________________

Notes To find the mean value of a function in interval [a ,b]we represent their sum by integrating the function between a and b, and represent the number of values as the interval b−a

f= 1b−a∫a

b

f (x )dx

WB B1 Find the exact mean value of a¿ f (x )= 4

√2+3 x over the interval [2,6 ]

FM Methods in calculus name _______________________

Transformations of functions and the mean value

f= 1b−a∫a

b

f (x )dx

If you add a fixed value to all the values of f(x), then the mean will go up by that amount

If you change the sign of all the numbers, then the mean will change sign

If you multiply all the numbers of f(x) by a scale factor, then the mean also gets multiplied by that scale factor

f ( x )+6 has mean value f +6 over the interval [a, b]

−f ( x ) has mean value −f over the interval [a, b]

5 f ( x ) has mean value 5× f over the interval [a, b]

FM Methods in calculus name _______________________

WB B2 f ( x )= 4

1+ex

a) Show that the mean value of f(x) over the interval [ ln 2 , ln 6 ] is 4 ln 9

7ln 3

b) Use (a) to find the mean value over the interval [ ln 2 , ln 6 ] of f ( x )+4c) Use geometric considerations to find the mean value of – f (x )over

[ ln 2 , ln 6 ]

FM Methods in calculus name _______________________

FM Methods in calculus name _______________________

Calculus and inverse functions

WB C1

a) show that ddx

(arcsin x )=¿ 1

√1−x2

b) show that ddx

(arccos x )=−¿ 1

√1−x2

c) show that ddx

(arc tan x )=¿ 11+ x2

FM Methods in calculus name _______________________

function Gradient function

y4=3 x

1y=1x

√ y=cos x

sin y=3 x2

x2+ y2=16

e y=e−3 x

ln y= ln x+5 x

FM Methods in calculus name _______________________

WB C2 Given y=arcsin x2, find

dydx

a) Using implicit differentiation

b) Using the chain rule and the formula for ddxarcsin x

WB C3 Given y=arctan( 1−x1+x )find  dydx

FM Methods in calculus name _______________________

FM Methods in calculus name _______________________

WB D1 Byusing an appropriate substitution,

a) show that ∫ 1√a2−x2

dx=arcsin ( xa )+C where a is a positive constant and |x|<a

Now show that b¿∫ 1a2+x2

dx=1aarctan( xa )+C

FM Methods in calculus name _______________________

WB D2 Find ∫ 4

5+x2dx

WB D3 Find ∫ 1

25+9 x2dx

FM Methods in calculus name _______________________

WB D4Find ∫

−√3/4

√3 /4 1√3−4 x2

dx

WB D5 Find ∫ x+4

√1−4 x2dx

FM Methods in calculus name _______________________

FM Methods in calculus name _______________________

WB D6 f ( x )= x+2x2+9

a) Show that ∫ f (x )dx=A ln (x2+9 )+Barctan ( x3 )+Cb) Show that the mean value of f(x) over the interval [0 ,3 ] is

16ln2+ 1

18π

c) Use (a) to find the mean value over the interval [0 ,3 ] of f ( x )+ lnk

Where k is a positive constant, giving your answer in the form p+ 16ln q

FM Methods in calculus name _______________________

FM Methods in calculus name _______________________

WB E1a) show that ∫ 1

a2−x2 dx= 1

2aln| a+xa−x|+C where a is a constant

FM Methods in calculus name _______________________

Notes partial fractions and quadratic factors

When a partial fraction includes a quadratic factor of the form x2+cwe need to put a linear numerator of the form Ax+B on the partial fraction

WB E2 5 x2+ x−10(x+1) (x2−3 )

= Ax+1

+ Bx+Cx2−3

find A, B and C

FM Methods in calculus name _______________________

WB E3 Show that ∫ 1+x

x3+9 xdx=A ln x2

x2+9+Barctan( x3 )+C where A and B are

constants to be found

FM Methods in calculus name _______________________

WB E4 a) express

x4+xx4+5x2+6

as partial fractions

b) Hence find ∫ x4+xx4+5 x2+6

dx

FM Methods in calculus name _______________________

WB E5a) express

x4+xx4+5 x2+6

as partial fractions

b) Hence find ∫ x4+xx4+5 x2+6

dx

FM Methods in calculus name _______________________