Game plan 0 · Units 3 and 4

The Mechanics of Methods

The other six game plans tell you which technique a question needs. This one is the techniques themselves. Every other game plan assumes you can do these without stopping to think.

Work on these all year. By the last two months before the external, you want them automatic, so that your thinking in the exam goes on the question and not on the algebra.

Part 1

Differentiation techniques

You need to know the standard derivatives by heart and choose the right rule for how they are put together.

Standard derivatives

$f(x)$$f'(x)$
$x^{n}$$nx^{n-1}$
$e^{x}$$e^{x}$
$\ln x$$\dfrac{1}{x}$
$\sin x$$\cos x$
$\cos x$$-\sin x$

Putting them together

Chain rule

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

Product rule

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

Quotient rule

$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$$
Game plan 0.1

Choosing a rule

  1. 1Is it a single standard function of $x$? Use the table and you are done.
  2. 2Is one function inside another, like $e^{3x^{2}}$ or $\sin(5x-1)$? Use the chain rule. Differentiate the outside, keep the inside, then multiply by the derivative of the inside.
  3. 3Are two functions of $x$ multiplied together? Use the product rule. Write down $u$, $v$, $u'$ and $v'$ before you substitute.
  4. 4Is one function of $x$ divided by another? Use the quotient rule. The top starts with $v$, so the subtraction goes the right way round.
  5. 5Inside the product or quotient rule, does $u$ or $v$ need the chain rule? Do that first, on the side.
  6. 6Tidy up only as far as the question needs. Factorise if you will be setting the derivative to zero next.

Watch for

The two slips I see most are dropping the derivative of the inside and losing the minus sign when you differentiate $\cos x$.

In the Topic Guides

Part 2

Integration techniques

Integration is differentiation run backwards. Most of the work is getting the expression into a form you can integrate one term at a time.

Standard integrals

$$\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c,\quad n\ne-1$$
$$\int (ax+b)^{n}\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+c$$
$$\int \frac{1}{x}\,dx=\ln x+c,\quad x>0$$
$$\int \frac{1}{ax+b}\,dx=\frac{1}{a}\ln(ax+b)+c$$
$$\int e^{ax+b}\,dx=\frac{1}{a}e^{ax+b}+c$$
$$\int \sin(ax+b)\,dx=-\frac{1}{a}\cos(ax+b)+c$$
$$\int \cos(ax+b)\,dx=\frac{1}{a}\sin(ax+b)+c$$
$$\int_{a}^{b}f(x)\,dx=F(b)-F(a)$$
Game plan 0.2

Integrating

  1. 1Rewrite first. Turn roots and fractions into powers of $x$, expand brackets and split fractions over a single term.
  2. 2Is there a linear expression $ax+b$ inside? Integrate as normal, then divide by $a$.
  3. 3Integrate one term at a time and add $+c$ to an indefinite integral.
  4. 4Have you been given a data point, like $f(0)=3$? Substitute it to find $c$.
  5. 5Is it a definite integral? Write the antiderivative in square brackets, then substitute the top bound and subtract the bottom one. You can leave $+c$ off.
  6. 6Check by differentiating your answer. You should get back what you started with.

Watch for

With a linear expression inside, you divide by $a$. If you multiply by it, you have done the chain rule forwards by mistake.

In the Topic Guides

Part 3

Finding the equation of a graph

Here you are given the graph and need its equation. Every parameter in the general form shows up as a feature you can read off the graph, so the work is matching each feature to its parameter.

What each parameter tells you

General formWhere to find each parameter
Exponential$y=ae^{bx}+k$$k$ is the horizontal asymptote. The $y$-intercept is $a+k$, so it gives you $a$. One more point gives you $b$.
Logarithmic$y=a\ln(x-h)+k$$h$ is the vertical asymptote. At $x=h+1$ the log is zero, so $y=k$ there. One more point gives you $a$.
Sinusoidal$y=a\sin(b(x-h))+k$$k$ is the centre line, halfway between the maximum and minimum. $|a|$ is half the distance from the maximum to the minimum. Read the period off the graph and use $b=\frac{2\pi}{\text{period}}$. $h$ is where the curve crosses the centre line going up. The same goes for cosine, but $h$ is where it reaches a maximum.
xy(0, 3)y = 1
Exponential · $y=2e^{x}+1$Asymptote $y=1$ gives $k=1$. The $y$-intercept is $a+k=3$, so $a=2$.
xy(2, 2)x = 1
Logarithmic · $y=2\ln(x-1)+2$Asymptote $x=1$ gives $h=1$. At $x=h+1=2$, $y=k=2$.
xy(1, 3)(3, −1)y = 1
Sinusoidal · $y=2\sin\left(\frac{\pi}{2}x\right)+1$Centre line $y=1$ and amplitude $2$. The period is 4, so $b=\frac{2\pi}{4}=\frac{\pi}{2}$.
Game plan 0.3

Finding the equation of a graph

  1. 1What type of function is it? A horizontal asymptote means exponential, a vertical asymptote means log, and a repeating wave means sine or cosine. Write down the general form.
  2. 2Read the asymptote or the centre line. That gives you $k$, or $h$ for a log.
  3. 3For a trig graph, read the maximum, the minimum and the period. Use them to find $a$ and $b$.
  4. 4Count the unknowns that are left. Substitute one point from the graph for each one, starting with the intercepts.
  5. 5If two unknowns are left, use two points and solve the equations simultaneously. For an exponential, dividing one equation by the other often removes $a$.
  6. 6Check your equation with a point you did not use, and check the direction. A negative $a$ flips the graph.

Watch for

Read the scale on the axes before you read any values. Trig graphs are often marked in multiples of $\frac{\pi}{4}$ or in hours, and a misread period throws off $b$.

In the Topic Guides

Part 4

The unit circle

Technology-free questions expect you to know these exact values. They come up in derivatives, definite integrals and the sine and cosine rules.

Exact values

$\theta$$\sin\theta$$\cos\theta$$\tan\theta$
$0$$0$$1$$0$
$\frac{\pi}{6}$$\frac{1}{2}$$\frac{\sqrt3}{2}$$\frac{1}{\sqrt3}$
$\frac{\pi}{4}$$\frac{1}{\sqrt2}$$\frac{1}{\sqrt2}$$1$
$\frac{\pi}{3}$$\frac{\sqrt3}{2}$$\frac{1}{2}$$\sqrt3$
$\frac{\pi}{2}$$1$$0$undefined

Which ratios are positive

All Sin Tan Cos 0 π/2 π 3π/2

Going anticlockwise from the first quadrant: All Stations To Central.

Game plan 0.4

Solving a trig equation exactly

  1. 1Rearrange so the trig ratio is on its own, for example $\sin x=-\frac{1}{2}$.
  2. 2Ignore the sign and find the reference angle from the table. Here it is $\frac{\pi}{6}$.
  3. 3Use the sign to pick the quadrants. Sine is negative in the third and fourth quadrants.
  4. 4Write the angle in each quadrant: $\pi+\frac{\pi}{6}=\frac{7\pi}{6}$ and $2\pi-\frac{\pi}{6}=\frac{11\pi}{6}$.
  5. 5Check the domain. Add or subtract whole periods until you have every solution in it, and no more.

In the Topic Guides

Next: Game plan 1, Calculus. Back to the External Game plan.

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