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A Level Maths > Equations To Remember > Flashcards

Equations To Remember Flashcards

1
Q

How to integrate f(ax + b)

A
  • Consider the value which would differentiate to give the original function
  • Workout the coefficient needed to balance the differentiated function and the function that you need
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2
Q

What is dy/dx of e^5x-4

A

5e^5x-4

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3
Q

How to integrate k(f’(x))/(f(x))

A
  • Try ln|f(x)| and differentiate to check

- Adjust any constants to match the original function

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4
Q

How to integrate k f’(x)*f(x)^n

A
  • Try (f(x))^n+1 and differentiate to check

- Adjust any constants to match original function

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5
Q

Cosine rule

A

Cos C = (a^2 +b^2 -c^2)/2ab

Or a^2 = b^2 + c^2 - 2bcCosA

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6
Q

On a graph y=x(x+2)^2(3+x)

Where does it touch the x-axis and where does it cross the x-axis

A

Touches at x= -2

Crosses at x= -3

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7
Q

What is the y-intercept of y = 4^x

A

1

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8
Q

What is the y-intercept of y= 2e^x

A

2

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9
Q

How to convert y = ax^n into y = mx + c

A 
y = ax^n
Log(y) = Log(a) + Log(x^n) 
Log(y) = Log(a) + nLog(x)
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10
Q

How to solve fg(x)

A 
Solve g(x) to make x the subject
Place into function of f(x)
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11
Q

Graph of y= |f(x)|

A

Reflect lines below x-axis above the x-axis

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12
Q

Graph of y=f(|x|)

A

Mirror graph in y-axis showing reflection of positive values

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13
Q

Nth term of arithmetic sequence

A

Un = a + (n-1)d

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14
Q

Nth term of geometric sequence

A

Un = ar^(n-1)

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15
Q

Sum of first n terms for an arithmetic series

A

Sn = n/2(2a+(n-1)d)
or
Sn = n/2(a+l)

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16
Q

Sum of first n terms for geometric series

A

Sn = a(1-r^n)/1-r
or
Sn = a(r^n-1)/r-1

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17
Q

Sum to infinity for a convergent series

A

S = a/1-r

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18
Q

Binomial expansion when n is a fraction or negative

A

(1+x)^2 = 1 + nx + (n(n-1)/2!)x^2 + (n(n-1)(n-2)/3!)x^3

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19
Q

How to use binomial expansion on (a+bx)^n

A

(a+bx)^n = (a(1+(b/a)x)^n = a^n(1+(b/a)x)^n

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20
Q

Area of a sector in radians

A

(1/2)r^Ø

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21
Q

small angle approximation of cosØ

A

CosØ = 1 - (ø^2)/2

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22
Q

1 + tan^2x =

A

Sec^2x

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23
Q

1 + cot^2x

A

Cosec^2x

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24
Q

Sin(A+B)

A

sinAcosB + cosAsinB

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25
Q

Cos(A+B)

A

cosAcosB - sinAsinB

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26
Q

Tan(A+B)

A

(tanA + tanB)/(1 - tanAtanB)

27
Q

Sin2A

A

2sinAcosA

28
Q

Cos2A

A 
Cos^2A - Sin^2A
or
2cos^2A - 1
or 
1 - 2sin^2A
29
Q

Tan2A

A

2tanA/1-tan^2A

30
Q

asinx + bcosx can be expressed as …

A

Rsin(x+ã)
where Rcosã = a and Rsinã = b
And R = sqrt(a^2 + b^2)

31
Q

acosx + bsinx can be expressed as …

A

Rcos(x-ã)
where Rcosã = b and Rsinã = a
And R = sqrt(a^2 + b^2)

32
Q

How to convert between parametric and Cartesian equation

A

Solve one of the parametric equations to find x

Substitute into the other parametric equation

33
Q

Domain of f(x) for parametric equations

A

Range of p(t)

Where x = p(t)

34
Q

Range of f(x) for a parametric equation

A

Range of q(t)

Where y = q(t)

35
Q

Chain rule:

A

dy/dx = dy/du x du/dx

36
Q

Product rule:

A

uv’ + vu’

37
Q

Quotient rule:

A

vu’-uv’/v^2

38
Q

Derivative of arcsinx

A

1/sqrt(1-x^2)

39
Q

Derivative of arccosx

A

-1/sqrt(1-x^2)

40
Q

Derivative of arctanx

A

1/(1+x^2)

41
Q

How to differentiate implicitly

A
  • Differentiate x terms as normal
  • For terms with y, differentiate as you would with x and then multiply by dy/dx
  • d/dx of xy = x(dy/dx) + y
  • Factor out dy/dx
42
Q

What does it tell you about the function if f’’(x) is < or > than 0

A

If f’’ < 0, function is concave

If f’’ > 0, function is convex

43
Q

Newton-Raphson formula:

A

X(n+1) = Xn - f(Xn)/f’(Xn)

44
Q

Integrate 1/x

A

ln|x| + c

45
Q

Integrate cosecx•cotx

A

-cosecx + c

46
Q

Integrate cosec^2x

A

-cotx + c

47
Q

Integrate sec^2x

In formula booklet

A

tanx + c

48
Q

Integrate secx•tanx

A

secx + c

49
Q

Reverse chain rule:

Integrate f’(ax+b)

A

1/a•f(ax+b) +c

50
Q

Integrate k(f’(x)/f(x))

A

try ln|f(x)| and differentiate to check and then adjust any constants

51
Q

Integrate k(f’(x)(f(x))^n)

A

Try (f(x))^(n+1) and differentiate to check, and then adjust any constants

52
Q

Integration by parts:

E.g, integrate uv’ (u•dv/dx)

A 
uv’ = uv - integral of (vu’)
u•dv/dx = uv - integral of v•du/dx
53
Q

dy/dx = f(x)g(y) can also be written as

A

Integral of (1/g(y)) = integral of f(x)

54
Q

Prove that sqrt(2) is irrational

A 
*write assumption*
then sqrt(2) = a/b
And assume that a/b is a fraction in its simplest form 
2 = a^2/b^2
a^2 = 2b^2
This means that a^2 is even 
So (2n)^2 = b^2
This means that b^2 is even 
If a and b are both even they have a common factor of 2. This contradicts the statement that a and b have no common factors so sqrt(2) is irrational
55
Q

How to prove that there is an infinite number of prime numbers

A

assumption
List all the prime numbers: p1, p2, p3, …, pn
Consider the number N = p1•p2•p3• … •pn + 1
You can divide n by any of the prime numbers in the equation as you get a remainder of 1 so N has no factors.
N must either be prime or have a prime factor which isn’t on the list of possible prime numbers
Therefore there is an infinite number of prime numbers

56
Q

What is the discriminant and what does it show about the roots of an equation?

A

If b^2 - (4ac) > 0, then f(x) has two roots
If b^2 - (4ac) = 0, then f(x) has one repeated root
If b^2 - (4ac) < 0, then f(x) has no real roots

57
Q

What is the set notation for x>-2 and x<4?

A

{x: -2

58
Q

What is the set notation for x3?

A

{x: x3}

59
Q

What does the graph of y=k/x look like

A

If k>0, two curves approaching asymptotes in diagonal quadrants, (+,+) and (-,-)
If k<0, two curves approaching asymptotes in diagonal quadrants, (+,-) and (-,+)

60
Q

What does the graph of y=k/x^2 look like

A

If k>0, two curves approaching asymptotes in adjacent quadrants, all above the x-axis
(conductors graph)
If k<0, two curves approaching asymptotes in adjacent quadrants, all below the x-axis

61
Q

If y=a^(kx), where k is real and a>0, then what is dy/dx?

A

dy/dx = ka^(kx)ln(a)

62
Q

How to find the gradient of a point on a curve given parametric equations

A
  • Differentiate each equation without changing into Cartesian equation.
  • Divide dy/dt by dx/dt to get dy/dx.
  • Substitute in a a value of t to get the gradient
63
Q

Solving differential equations

A

Differentiate implicitly
Factor out dy/dx

Separate the variables
Integrate both sides

A Level Maths (8 decks)

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