A-level Mathematics/AQA/MPC3

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Functions[edit]

Domain and range of a function[edit]

In general:

  •  f(x) is called the image of  x .
  • The set of permitted  x values is called the domain of the function
  • The set of all images is called the range of the function

Modulus function[edit]

The modulus of  x , written  |x| , is defined as

 |x| =  \begin{cases} x & \mbox{for } x \ge 0 \\ -x & \mbox{for } x < 0 \end{cases}

Differentiation[edit]

Chain rule[edit]

The chain rule states that:

If  y is a function of  u , and  u is a function of  x ,

 \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}

Product rule[edit]

The product rule states that:

If  y = uv , where  u and  v are both functions of  x , then

 \frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx}

An alternative way of writing the product rule is:

 (uv)' = uv' + u'v \,\!

Quotient rule[edit]

The quotient rule states that:

If  y = \frac{u}{v} , where  u and  v are functions of  x , then

 \frac{d}{dx} \left ( \frac{u}{v} \right ) = \frac{ v \frac{du}{dx} - u \frac{dv}{dx} }{v^2}

An alternative way of writing the quotient rule is:

 \left ( \frac{u}{v} \right )' = \frac{u'v - uv'}{v^2}

x as a function of y[edit]

In general,

 \frac{dy}{dx} = \frac{1}{ \frac{dx}{dy} }

Trigonometric functions[edit]

The functions cosec θ, sec θ and cot θ[edit]

 \operatorname{cosec}{\theta} = \frac{1}{\sin{\theta}}


 \sec{\theta} = \frac{1}{\cos{\theta}}


 \cot{\theta} = \frac{1}{\tan{\theta}}

Standard trigonometric identities[edit]

 \cot{\theta} = \frac{ \cos{ \theta } } {\sin{ \theta } }


 \sec^2{\theta} = 1 + \tan^2{\theta} \,\!


 \operatorname{cosec}^2{\theta} = 1 + \cot^2{\theta}

Differentiation of sin x, cos x and tan x[edit]

 \frac{d}{dx} \left ( \sin{x} \right ) = \cos{x}


 \frac{d}{dx} \left ( \cos{x} \right ) = -\sin{x}


 \frac{d}{dx} \left ( \tan{x} \right ) = \sec^2{x}

Integration of sin(kx) and cos(kx)[edit]

In general,

 \int \cos{kx}\ dx = \frac{1}{k} \sin{kx} + c


 \int \sin{kx}\ dx = - \frac{1}{k} \cos{kx} + c

Exponentials and logarithms[edit]

Differentiating exponentials and logarithms[edit]

In general,

 \mbox{when}\ y = e^{kx},\ \frac{dy}{dx} = ke^{kx}


 \int e^{kx}\ dx = \frac{1}{k} e^{kx} + c

Natural logarithms[edit]

If y=\ln{x}, then

 \frac{dy}{dx} = \frac{1}{x}


It follows from this result that

 \int \frac{1}{x}\ dx = \ln{x} + c


 \int \frac{f'(x)}{f(x)}\ dx = \ln{f(x)} + c,\ \mbox{provided}\ f(x) > 0

Integration[edit]

Integration by parts[edit]

 \int u \frac{dv}{dx}\ dx = uv - \int v \frac{du}{dx}\ dx

Standard integrals[edit]

 \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}{ \left ( \frac{x}{a} \right ) } + c


 \int \frac{dx}{\sqrt{(a^2 - x^2)}} = \sin^{-1}{ \left ( \frac{x}{a} \right ) } + c

Volumes of revolution[edit]

The volume of the solid formed when the area under the curve  y = f(x) , between  x = a and  x = b , is rotated through 360° about the  x -axis is given by:

 V = \pi \int_a^b y^2\ dx


The volume of the solid formed when the area under the curve  y = f(x) , between  y = a and  y = b , is rotated through 360° about the  y -axis is given by:

 V = \pi \int_a^b x^2\ dy

Numerical methods[edit]

Iterative methods[edit]

An iterative method is a process that is repeated to produce a sequence of approximations to the required solution.

Numerical integration[edit]

Mid ordinate rule

\int_a^b y\ dx \approx h \lbrack y_{\frac{1}{2}} + y_{\frac{3}{2}} + \ldots +  y_{n-\frac{3}{2}} +  y_{n-\frac{1}{2}} \rbrack


 \mbox{where}\ h = \frac{b-a}{n}


Simpson's rule

\int_a^b y\ dx \approx \frac{h}{3} \lbrack \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 )  \rbrack


\mbox{where}\ h = \frac{b-a}{n}\ \mbox{and}\ n\ \mbox{is even}