Post on 06-Sep-2019
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 _______________________