A-level Mathematics/OCR/C3/Formulae

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By the end of this module you will be expected to have learnt the following formulae:

Transformations of Graphs[edit]

Reflection[edit]

  1. y = -f \left (x \right )\, is a reflection of y = f \left (x \right )\, through the x axis.
  2. y = f \left (-x \right )\, is a reflection of y = f \left (x \right )\, through the y axis.
  3. y = \begin{vmatrix}f\left(x\right)\end{vmatrix} is a reflection of y = f \left (x \right )\, when y < 0, through the x-axis.
  4. y =f\left(\begin{vmatrix}x\end{vmatrix}\right) is a reflection of y = f \left (x \right )\, when x < 0, through the y-axis.
  5. y = f^{-1} \left (x \right )\, is a reflection of y = f \left (x \right )\, through the line y = x.
    Note: f^{-1} \left (x \right ) exists only if f \left (x \right ) is bijective, that is, one-to-one and onto.

Stretching[edit]

  1. y = af \left (x \right )\, is stretched toward the x-axis if 0 < a < 1\, and stretched away from the x-axis if a > 1\,. In both cases the change is by a units.
  2. y = f \left (bx \right )\, is stretched away from the y-axis if 0 < b < 1\, and stretched toward the y-axis if b > 1\,. In both cases the change is by b units.

Translations[edit]

  1. y = f \left (x - h \right )\, is a translation of f(x) by h units to the right.
  2. y = f \left (x + h \right )\, is a translation of f(x) by h units to the left.
  3. y = f \left (x \right ) + k\, is a translation of f(x) by k units upwards.
  4. y = f \left (x \right ) - k\, is a translation of f(x) by k units downwards.

Natural Functions[edit]

  1. e^{\ln x} = \ln e^x = x\,
  2. y\left(t\right)=y_0e^{kt}\,, where y(t) is the final value, y_0 is the initial value, k is the growth constant, t is the elapsed time.
  3. k = - \frac {\ln 2}{half-life}, k for calculations involving half-lives.

Trigonometry[edit]

Reciprocal Trigonometric Functions and their Inverses[edit]

  •  \sec \theta \equiv \frac{1}{\cos \theta}
  •  \operatorname{cosec}\ \theta \equiv \frac{1}{\sin \theta}
  •  \cot \theta \equiv \frac{1}{\tan \theta}\equiv \frac{\cos \theta}{\sin \theta}
  •  \sec ^2 \theta \equiv 1 + \tan ^2 \theta
  •  \operatorname{cosec} ^2\ \theta \equiv 1 + \cot ^2 \theta

Angle Sum and Difference Identities[edit]

  • \sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)\,
  • \cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B)\,
  • \tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}

Note: The sign \mp means that if you add the angles (A+B) then you subtract in the identity and vice versa. It is present in the cosine identity and the denominator of the tangent identity.

Double Angle Identities[edit]

  •  \sin 2A \equiv 2 \sin A \cos A
  •  \cos 2A \equiv \cos ^2 A - \sin ^2 A \equiv 1 - 2\sin ^2 A \equiv 2\cos ^2 A - 1
  •  \tan 2A \equiv \frac{2 \tan A}{1 - \tan ^2 A}

Combination of Trigonometric Functions[edit]

Using radians r = amplitute α = phase.

r = \sqrt{a^2+b^2}

a\sin x+b\cos x=r\cdot\sin(x+\alpha)\,

where

\alpha = \arcsin\frac{b}{r}

a\sin x+b\cos x=r\cdot\cos(x-\alpha)\,

where

\alpha = \arccos\frac{b}{r}

Differentiation[edit]

  • If  y = \operatorname{e}^{kx}\,, then  \frac{dy}{dx} = k\operatorname{e}^{kx}
  • If  y = \ln x\,, then  \frac{dy}{dx} = \frac{1}{x}
  • If  y = f(x).g(x)\,, then  \frac{dy}{dx} = f^'(x)g(x) + g^'(x)f(x)
  • If  y = \frac{f(x)}{g(x)}, then  \frac{dy}{dx} = \frac{f^'(x)g(x) -g^'(x)f(x)}{\left\{g(x)\right\}^2}
  •  \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}
  • If  y = f[g(x)]\,, then  \frac{dy}{dx} = f^'[g(x)].g^'(x)
  •  \frac{dy}{dt} = \frac{dy}{dx}.\frac{dx}{dt}

Integration[edit]

  •  \int \operatorname{e}^{kx}\, dx = \frac{1}{k}\operatorname{e}^{kx} + c
  •  \int \frac{1}{x}\, dx = \ln \left|x\right| + c

For volumes of revolution:

  •  V_x = \pi \int_{a}^{b} y^2\, dx
  •  V_y = \pi \int_{c}^{d} x^2\, dy

Numerical Methods[edit]

Simpson's Rule

\int^b_a y dx \approx \frac{1}{3}h\left\{\left(y_0 + y_n\right) + 4\left(y_1 + y_3 + \ldots + y_{n-1}\right) + 2\left(y_2 + y_4 + \ldots + y_{n-2}\right) \right\}

whereh = \frac{b-a}{n} and n is even

This is part of the C3 (Core Mathematics 3) module of the A-level Mathematics text.