Pure 2 Flashcards
How to prove by contradiction?
Start by assuming it’s not true. Use steps to lead to something impossible, a contradiction. Conclude the assumption it’s not true is incorrect.
In proofs, how can rational and irrational numbers be differentiated?
Rational numbers can be expressed like b where and b are integers, irrational numbers cannot.
What is Q?
The set of all rational numbers.
What does the modulus of a number do?
Make it not negative
When f(x) >= 0, the modulus of f(x) =
f(x)
When f(x) < 0, the modulus of f(x) =
-f(x)
How to sketch y = modulus(ax+b)?
Sketch y = ax+b, then reflect the part of the graph below the x-axis in the x-axis
What is domain and range?
Domain = all possible inputs, Range = all possible outputs
A mapping is a function if every input has a distinct output, functions can be what or what?
One-to-one or many-to-one
What type of function is fg(x)?
Composite
What is the inverse of f(x)?
f⁻¹(x)
What does y=f(x) look like in relation to y=f⁻¹(x)?
A reflection in the line y=x
How do the domains and ranges of inverse functions relate?
The domain of f(x) is the range of f⁻¹(x). The range of f(x) is the domain of f⁻¹(x).
How to sketch y = modulus(f(x))?
Sketch f(x), reflect any parts under the x-axis in the x-axis, delete all parts under the x-axis
How to sketch y = f(modulus(x))?
Sketch f(x) for x >= 0, reflect this in the y-axis
What does f(x+a) do?
Translation left by a
What does f(x) + a do?
Translation up by a
What does f(-x) do?
Reflect in y-axis
What does -f(x) do?
Reflect in x-axis
What does f(ax) do?
Horizontal stretch of scale factor 1/a
What does af(x) do?
Vertical stretch of scale factor a
Formula for nth term of an arithmetic sequence
Un = a + (n-1)d; a is the first term and d is the common difference
Formula for the sum of the first n terms of an arithmetic sequence
Sn = (n/2)(2a + (n-1)d); a is the first term and d is the common difference
Formula for the nth term of a geometric sequence
Un = ar^(n-1); a is the first term and r is the common ratio
Formula for the sum of the first n terms of a geometric series
Sn = (a(1-r^n))/(1-r), where r ≠ 1; a is the first term and r is the common ratio
What is a convergent series?
A geometric series where
Formula for the sum to infinity of a convergent geometric series
S∞ = a/(1-r)
Explain sigma notation
Greek capital letter sigma. Bottom shows the value of the variable to start on. Top shows the value of the variable to end on. Right shows the series with respect to the variable.
What does a recurrence relation do? With general rule
Defines each term of a sequence as a function of the previous term. For example, Un+1 = f(Un)
Expand (1+x)^n
1 + nx + ((n(n-1))/2!)x² + ((n(n-1)(n-2))/3!)x³ + …
Expand (a+bx)^n
(a^n)(1+(b/a)x)^n
When is the expansion of (1+x)^n valid?
When modulus(bx)<1 or modulus(x) < 1/modulus(b)
When is the expansion of (a+bx)^n valid?
When modulus((b/a)/x) < 1 or modulus(x) < modulus(a/b)
What is a radian?
A measure of angles. 1 radian = the angle AOB of the radius of a segment with equal arc length and radius
What is 2π radians in degrees?
360
What is π radians in degrees?
180
What is 1 radian in degrees?
180/π
Convert 30° to radians
π/6
Convert 45° to radians
π/4
Convert 60° to radians
π/3
Convert 90° to radians
π/2
Convert 180° to radians
π
Convert 360° to radians
2π
sin(π/6)
1/2
sin(π/3)
√3/2
sin(π/4)
√2/2
cos(π/6)
√3/2
cos(π/3)
1/2
cos(π/4)
√2/2
tan(π/6)
√3/3
tan(π/3)
√3
tan(π/4)
1
sin(π - θ)
sin(θ)
sin(π + θ)
-sin(θ)
sin(2π - θ)
-sin(θ)
cos(π - θ)
-cos(θ)
cos(π + θ)
-cos(θ)
cos(2π - θ)
cos(θ)
tan(π - θ)
-tan(θ)
tan(π + θ)
tan(θ)
tan(2π - θ)
-tan(θ)
Arc length formula in radians
l = rθ
Area of a sector in radians
A = (1/2)r²θ
When θ is small and in radians, sin(θ) ≈
θ
When θ is small and in radians, tan(θ) ≈
θ
When θ is small and in radians, cos(θ) ≈
1 - (θ²/2)
Which equation links sec and cos?
sec = 1/cos
Which equation links cosec and sin?
cosec = 1/sin
Which equation links tan and cot?
cot = 1/tan
Which equation links sin, cos, and cot?
cot = cos/sin
Which equation links sin and cos?
sin² + cos² = 1
Which equation links tan and sec?
1 + tan² = sec²
Which equation links cot and cosec?
1 + cot² = cosec²
Expand sin(A + B)
sinAcosB + cosAsinB
Expand sin(A - B)
sinAcosB - cosAsinB
Expand cos(A + B)
cosAcosB - sinAsinB
Expand cos(A - B)
cosAcosB + sinAsinB
Expand tan(A + B)
(tanA + tanB) / (1 - tanAtanB)
Expand tan(A - B)
(tanA - tanB) / (1 + tanAtanB)
sin(2A) =
2sinAcosA
cos(2A) =
cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
tan(2A) =
(2tanA) / (1 - tan²A)
For R Sin or Cos (x ± α), what does R =
√(a² + b²)
For the parametric equations x = p(t) and y = q(t), with the Cartesian equation y = f(x), how are domains and ranges linked?
The domain of f(x) is the range of p(t), the range of f(x) is the range of q(t)
Differentiate sin(kx)
k cos(kx)
Differentiate cos(kx)
-k sin(kx)
Differentiate e^(kx)
k e^(kx)
Differentiate ln(x)
1/x
Differentiate a^(kx)
(a^(kx))(k)(ln(a))
What is the chain rule?
dy/dx = (dy/du) × (du/dx)
How to get from dx/dy to dy/dx?
dy/dx = 1 / (dx/dy)
What is the product rule?
If y = uv, then dy/dx = u(dv/dx) + v(du/dx)
What is the quotient rule?
If y = u/v, then dy/dx = (v(du/dx) - u(dv/dx)) / v²
Differentiate tan(kx)
k sec²(kx)
Differentiate cosec(kx)
-k cosec(kx) cot(kx)
Differentiate sec(kx)
k sec(kx) tan(kx)
Differentiate cot(kx)
-k cosec²(kx)
For parametric functions, how do you get dy/dx?
(dy/dt) / (dx/dt)
How to do implicit differentiation?
Differentiate y like x, then multiply it by dy/dx
f(x) is concave if f’‘(x) is what?
≤ 0
f(x) is convex if f’‘(x) is what?
≥ 0
What is the point called where a curve changes from concave to convex or vice versa?
The point of inflection
How is the point of inflection seen mathematically?
f’‘(x) changes signs
If the function f(x) is continuous on the interval [a, b] and f(a) and f(b) have opposite signs, then what?
f(x) has at least one root, x, which satisfies a
To determine a root to a given degree, use 1.763 as an example
If there is a change in sign for the upper and lower bound, e.g., 1.7635 and 1.7625 have a sign change
How can roots converge?
Staircase or spiderweb
What is the Newton-Raphson formula?
X(n+1) = X(n) - (f(X(n)) / f’(X(n)))
Integration of x^n
(x^(n+1)) / (n+1)
Integration of e^x
e^x
Integration of 1/x
ln(x)
Integration of cos(x)
sin(x)
Integration of sin(x)
-cos(x)
Integration of sec²(x)
tan(x)
Integration of cosec(x)cot(x)
-cosec(x)
Integration of cosec²(x)
-cot(x)
Integration of sec(x)tan(x)
sec(x)
Integration of f’(ax + b)
(1/a) * f(ax + b)
Formula for integration by parts
∫u * (dv/dx) = uv - ∫v * (du/dx)
Trapezium rule formula
∫y between a and b = (h/2)(y₀ + 2(y₁ + y₂ + y₃ … + Yn₋₁) + yn)
When dy/dx = f(x)g(y), how can this be rewritten?
∫(1/g(y)) = ∫f(x)
Integration of the limit of a sum for the integration of f(x) between a and b
limit of dx → 0 for ∑f(x) when x = a and b on the top
Distance of the origin to the point (x, y, z)
√(x² + y² + z²)
Distance between the points (x₁, y₁, z₁) and (x₂, y₂, z₂)
√((x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²)
How are unit vectors on the 3D axes denoted?
i, j, and k for x, y, and z respectively
If the vector a = xi + yj + zk makes an angle θ with the positive x-axis, how do you find it?
cos(θₓ) = x / modulus(a)
