Pure Maths Year 2 Flashcards
What’s a term to term rule example?
U1 = 2 U(n+1) = Un + 6
What’s a position to term rule example?
Un = 3n - 7
How do you find a limit for a convergent sequence?
Let U(n+1) = Un = L Then solve for L
e.g L= 0.5L +5
What’s the nth term for an arithmetic sequence?
Un = a + (n-1)d
What’s the formula for arithmetic series?
In the formula book
What’s the formula for geometric sequences?
Un = ar ^(n-1)
What’s the formula for geometric series?
In formula book
What’s the formula for infinite geometric series?
In formula book
How would you prove a sequence is increasing?
Show Un+1 - Un > 0
What does Sn - Sn-1 equal?
Un
How do you test if a mapping is a function?
Vertical line test
many-one or one-one
What’s the domain?
Input values
e.g x > 2 or x E R
What’s the range?
Possible outputs of a function
e.g f(x) < 6
What functions have an inverse
One-one functions only
How do you sketch a modulus function?
Sketch the graph and then reflect any parts below the x axis in the x axis
Write the intervals out for these modulus inequalities
|x-a|<b>b |x-2|>6<6</b>
- b < x-a < b
- 4 < x -3 < 4
x-a > b or x-a < -b
x-2 > 6 or x-2 < -6
What’s the factor theorem?
If f(3) = 0 then (x-3) is a factor
What’s an extension on the factor theorem?
If f(3/4) = 0 then (4x-3) is a factor
How do you find a quotient and remainder?
Divide by the denominator for quotient and remainder over denominator is what’s left
What must you do for a repeated factor in partial fractions?
A B C
——— + ——— + ———
ax + b cx + d (cx + d)²
What’s the formula for general binomial expansion?
In formula book
When is general binomial expansion valid?
|x|<1
Describe the graph
y=a sin(bx) and y=a cos(bx)
Amplitude is a
Period is 2π/b
Length of an arc
l=rθ
Area of a sector
A =1/2 r²θ
Where do you find small angle approximations
Formula book
Sin (A+B)
sinA cosB + cosA sinB
Cos(A+B)
cosA cosB - sinA sinB
tan (A+B)
In formula book
sin(2A)
2sinAcosA
cos(2A)
cos²(A) -sin²(A)
2cos²(A) -1
1-2sin²(A)
tan(2A)
2tan(A)
—————
1- tan²(A)
asinx + b cosx
Rewrite as R sin(x+a) and expand the brackets
Equate coefficients to get Rsina and Rcosa
Use Pythagoras for R
For a do tana = sinA/cosA
Sec
Cos^-1
Cosec
Sin^-1
Sec²(x)
Sec²(x)=1+tan²(x)
Cosec²(x)
Cosec²(x)=1+cot²(x)
What do you get if you differentiate e^x?
e^x
What do you get if you differentiate ln x
1/x
What’s the sin cos ladder?
S
C
-S
-C
What do you get if you differentiate tan x?
sec²x
What’s the chain rule?
dy dy du
— = — + —
dx du dx
What do you get if you differentiate inverse trig functions?
The formula book
What’s the product rule?
dy dv du
— = — U + — V
dx dx dx
What’s the quotient rule?
In formula book
How do you differentiate implicitly?
Differentiate with respect to y then multiply by dy/dx
What do you get if you differentiate a^x?
a^x ln a
You can prove this by writing ln y = xln a and differentiating implicitly
What methods can you use for integration?
Integration by substitution
Integration by parts - in formula book
What substitution would you use for ∫√a²-x² dx
x = a sinθ
What is ∫f’(x)/f(x) dx
ln |f(x)| + c
What is a concave and convex curve?
Concave: d²y/dx² < 0 /’’’’\
Convex: d²y/dx² < 0 .…/
What is a point of inflection?
d²y/dx² = 0
When does Newton Raphson Method fail?
If starting point is stationary point or too close to a stationary point
