Pure Year 2 Flashcards
What is an improper algebraic fraction?
One whose numerator has a degree equal to or larger than the denominator
What is a mapping? (Functions)
A function if every input has a distinct output.
What counts as a function and what doesn’t?
One-to-one functions
Many-to-one functions
But One-to-many is not a function!
What is the relationship between the graph of f(x) and f⁻¹(x)?
f⁻¹(x) is a reflection of f(x) in the line y=x
Describe the transformation f(x+a)
Horizontal translation of -a
Describe the transformation f(x) + a
Vertical translation of a
Describe the transformation f(ax)
Horizontal stretch of scale factor 1/a
Describe the transformation af(x)
Vertical stretch of scale factor a
Describe the transformation -f(x)
Reflects f(x) in the x-axis
Describe the transformation f(-x)
Reflects f(x) in the y-axis
What is the formula for the nth term of an arithmetic sequence?
uₙ = a + (n-1) d
What is the formula for the nth term of an geometric sequence?
uₙ = arⁿ⁻¹
What is the sum to infinity formula for a geometric series? What is the condition?
Series must be converging, |r|<1
S∞ = a /(1-r)
What is the sum of series formula for arithmetic series?
Sₙ = 0.5n(2a + (n-1)d)
or
Sₙ = 0.5n (a + l)
What is the sum of series formula for geometric series?
Sₙ = a(1-r ⁿ)/(1-r)
When is a sequence increasing?
If uₙ₊₁ > uₙ
When is a sequence decreasing?
If uₙ₊₁ < uₙ
When is a sequence periodic? What is the order of a periodic sequence?
If the terms repeat in a cycle. For a periodic sequence there is an integer k such that uₙ₊ₖ = uₙ for all n ε ℕ. The value k is the order of the sequence
When is the binomial expansion f (1+bx)ⁿ valid, when n is negative or a fraction?
|bx| < 1
or |x| < 1/ |b|
When is the binomial expansion f (a+bx)ⁿ valid, when n is negative or a fraction?
|ba/x| < 1
or |x| < |a/b|
What is a sector? A segment? (of a circle)
Sector is like a pizza slice
A segment is the area of the circle when the circle is cut by a chord
What is the formula for the area of a segment?
A = 0.5 r² (θ - sinθ)
What are the small angle approximations?
sinθ ≈ θ
tanθ ≈ θ
cosθ ≈ 1 - θ²/2
What does the graph of y = sec x look like? What is the domain and range? Period?
Symmetrical about y-axis
u and n shapes. Asymptotes at π/2, 3π/2 etc.
Domain: all real values of x. Except x ≠ π/2, 3π/2, … or any odd multiple of π/2
Range y ≤ -1, y ≥ 1
Period: 2π
What does the graph of y = cosec x look like? What is the domain and range? Period?
u and n shapes. Asymptotes at π, 2π etc.
Domain: all real values of x. Except x ≠ π, 2π, … or any integer multiple of π
Range y ≤ -1, y ≥ 1
Period: 2π
What does the graph of y = cot x look like? What is the domain and range? Period?
Period of π rads. Vertical asymptotes at 0, π, 2π etc.
Domain: All real values of x. Except x ≠ π, 2π, … or any integer multiple of π
Range: All real values of y
State the identity including tan and 1
1 + tan²x ≡ sec²x
State the identity including cot and 1
1 + cot²x ≡ cosec²x
What is the domain and range of y = arcsin x?
Domain: -1 ≤ x ≤ 1
Range: -π/2 ≤ arcsin x ≤ π/2
What is the domain and range of y = arccos x?
Domain: -1 ≤ x ≤ 1
Range: 0 ≤ arccos x ≤ π
What is the domain and range of y = arctan x?
Domain: all real values of x
Range: -π/2 ≤ arctan x ≤ π/2
What is the double angle formula for cos(2A)?
Cos(2A) ≡ cos²A - sin²A
≡ 2cos²A -1
≡ 1 - 2sin²A
What is the double angle formula for sin(2A)?
sin(2A) = 2sinAcosA
What is the double angle formula for tan(2A)?
2tanA/(1-tan²A)
What is the harmonic form of asinx ± bcosx?
R sin (x ± α) R > 0 0 < α < π/2 R cos α = a R sin α = b R = √(a² + b²)
What is the harmonic form of acosx ± bsinx?
R cos (x ∓ α)
What is the chain rule?
dy/dx = dy/du * du/dx
What is the product rule?
y=uv then
dy/dx = u dv/dx + v du/dx
What is the quotient rule?
y = u/v then dy/dx = (v du/dx - u dv/dx ) / v²
When is a function concave?
When f’‘(x) ≤ 0
When is a function convex?
When f’‘(x) ≥ 0
Where is a point of inflection?
A point at which f’‘(x) changes sign
Describe the conclusion when you have shown that there is a root in the interval [a, b]
If the function is CONTINUOUS on the interval [a, b] and f(a) and f(b) have opposite signs, then f(x) has at least one root x, which satisfies a < x < b
What is the Newton-Raphson formula?
xₙ₊₁ = xₙ - f(xₙ)/f’(xₙ)
What is the trapezium rule?
∫ᵇₐ y dx ≈ ½ h (y₀ + 2( y₁ + y₂ + … + yₙ₋₁) + yₙ)
h = (b-a)/n yᵢ = f(a + hᵢ)
How can you rewrite dy/dx = f(x)g(y) ?
∫ 1/g(y) dy = ∫ f(x) dx
If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θₓ with the positive x-axis, how do you calculate θₓ?
cos θₓ = x / |a|
If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θᵧ with the positive y-axis, how do you calculate θᵧ?
cos θᵧ = y / |a|
If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θ𝓏 with the positive z-axis, how do you calculate θ𝓏?
cos θ𝓏 = z / |a|
