Game plan 1: Calculus | Methods external game plans

Part 1

The two pathways

Decide first whether you are differentiating or integrating. Each pathway is a set of questions to ask yourself in order.

Game plan 1.1

Differentiation

1Check that you need to differentiate. The question will want a rate of change.
2What are you asked to do? Find a maximum, rate of change, velocity, gradient, or a tangent line.
3Is your equation ready to differentiate? If it is not, find the unknown first.
4How many times do you need to differentiate? Your answer to step 2 decides this.
5Which differentiation rules apply here? Game plan 0.1 will help you choose.
6What units is the question in? Give your answer in those units.

Game plan 1.2

Integration

1Check that you need to integrate. The question will give you a rate and want a total.
2You are told how something changes, but you need to find how much there is.
3What is your data point? The QCAA have to give you one so that you can find $c$.
4How many times do you need to integrate? Two data points can mean you integrate twice.
5Which integration rules apply here? Game plan 0.2 has them.
6What units is the question in? Give your answer in those units.

In the Topic Guides

Part 2

Applied calculus

These are the special cases where you use calculus to solve a problem. Each one has its own game plan.

Estimating area under curves
Area under and between curves
Turning points and points of inflection
Interpreting graphs with the first and second derivative
Optimisation
Total change of a function

Game plan 1.3

The trapezoidal rule: estimating area with strips

1Determine the width and number of strips.
2Determine each $x_n$.
3Determine each $f(x_n)$.
4Apply the trapezoidal rule.

Four strips of width $w$. The curve is concave up, so each strip pokes slightly above it, and the estimate is a little too big.

Strip width

$$w=\frac{b-a}{n}$$

Trapezoidal rule

$$A\approx\frac{w}{2}\big(\text{start}+2(\text{middle})+\text{end}\big)$$

Remember

If your curve is concave up, the trapezoidal rule will slightly overestimate the area. If it is concave down, it will slightly underestimate it.

Game plan 1.4

Area under a curve · Technology-free

1Sketch the function by hand and identify the area you need.
2Decide whether the area is above or below the $x$-axis. Fully above needs nothing extra. Fully below, take the absolute value of the integral. Partly both, split it and take the absolute value of the negative part.
3Identify the bounds from the given values, or the $x$-intercepts.
4Apply the Unit 3 Topic 4 integration techniques, then evaluate the definite integral.

Partly above, partly below. Split the integral at $x=2$ and take the absolute value of the part below.

Game plan 1.5

Area under a curve · Technology-active

1Sketch the function on your graphics calculator.
2Deal with any area below the axis the same way as Technology-free, using the absolute value.
3Find the bounds from the given values, or use the roots function for the $x$-intercepts.
4Use the gsolv, fint or mixed function to determine the area.

Tip

The mixed function on your graphics calculator can also evaluate an area that is partly above and partly below the axis.

Game plan 1.6

Area between curves

1Sketch both curves to see which is on top. Do this by hand if it is Technology-free, or on your calculator if it is Technology-active.
2Find the points of intersection. Set the functions equal if it is Technology-free, or use the intersection tool if it is Technology-active. These become your bounds.
3Integrate top minus bottom across those bounds.

The intersections give the bounds. Top minus bottom still works when the region dips below the $x$-axis.

Area between two curves

$$A=\int_{a}^{b}\big(f(x)-g(x)\big)\,dx$$

Watch for

In complex unfamiliar questions you may need to define one or both functions first. The QCAA also like to give you the area and ask you to find an unknown.

In the Topic Guides

Part 3

Analysing a function

Algebraic analysis of functions uses the first and second derivatives to find and describe the key features of a graph.

Game plan 1.7

Turning point and its nature

1Find the first derivative.
2Set $f'(x)=0$ to get the $x$-coordinate(s). Substitute back into $f(x)$ for the $y$-coordinate.
3Find the second derivative $f''(x)$.
4Substitute your $x$ into $f''(x)$. If $f''(x)>0$ it is concave up, so it is a minimum. If $f''(x)<0$ it is concave down, so it is a maximum.

Both points have $f'(x)=0$. The sign of $f''(x)$ tells them apart.

Game plan 1.8

Point of inflection

1Set $f''(x)=0$ to find the $x$-coordinate. Substitute back for $y$ if you need it.
2Use a table to check that $f''(x)$ changes sign. If it does not, it is not a point of inflection.

$f''(x)$ changes sign at the dot, so the concavity changes. That is what makes it a point of inflection.

Interpreting graphs with the first and second derivative

	$=0$	$>0$	$<0$
$f(x)$	$x$-intercept	Above the $x$-axis	Below the $x$-axis
$f'(x)$	Stationary point	$f$ increasing	$f$ decreasing
$f''(x)$	Potential point of inflection	$f$ concave up $\cup$	$f$ concave down $\cap$

Game plan 1.9

Reading a graph, step by step

1Start. What are you given, and what do you need to find?
2Mark the key $x$-values: intercepts, turning points and inflections.
3Translate. Apply the table above to each of those $x$-values.
4Join the dots using the signs: rising or falling, above or below, concave up or down.
5Check that the sign actually changed. A zero with no sign change is not a turning point or a point of inflection.

Game plan 1.10

Optimisation

1Identify the key information in the problem.
2Draw a picture.
3Construct an equation that represents the situation.
4Express your equation in terms of one variable.
5Set the first derivative to zero to find a potential turning point.
6Use the second derivative to check whether it is a minimum or a maximum.
7Answer the question in context.

Game plan 1.11

Total change of a function

1Identify that the function you are given is a rate function.
2Recognise that total change is found by integrating the rate function.
3Identify the bounds. For example, water released between 3 and 5 minutes means you integrate from 3 to 5.
4Evaluate the definite integral of the rate across those bounds.

In the Topic Guides

Part 4

Common QCAA language: Calculus

The wording of a question tells you which operation and which applied technique you need. Look for these phrases.

Core calculus language

Operation	What the question says
Differentiation	Rate of change, instantaneous rate of change, “the rate at which … is changing”, how fast … is changing. In kinematics, velocity is the first derivative and acceleration is the second. Gradient of the tangent or curve at a point. “Increasing or decreasing by [rate] per [unit].”Your move Look at the units in the answer. If they are a rate, like °C/hour or members/day, differentiate.
Integration	“$\int\ldots\,dx$”, “Determine $\int\ldots$”, evaluate the definite integral, antidifferentiate, antiderivative. Given the rate, find the quantity, for example velocity to displacement. “Initially …” with a rate means you find $c$. “Express your answer in the form $\ln(a)$” often signals a log antiderivative.Your move A rate goes in and a total comes out, so this is an integration question.

Operation

What the question says

Differentiation

Rate of change, instantaneous rate of change, “the rate at which … is changing”, how fast … is changing. In kinematics, velocity is the first derivative and acceleration is the second. Gradient of the tangent or curve at a point. “Increasing or decreasing by [rate] per [unit].”Your move Look at the units in the answer. If they are a rate, like °C/hour or members/day, differentiate.

Integration

“$\int\ldots\,dx$”, “Determine $\int\ldots$”, evaluate the definite integral, antidifferentiate, antiderivative. Given the rate, find the quantity, for example velocity to displacement. “Initially …” with a rate means you find $c$. “Express your answer in the form $\ln(a)$” often signals a log antiderivative.Your move A rate goes in and a total comes out, so this is an integration question.

Applied calculus language

Applied topic	What the question says
Estimating area under curves	“Estimate the area … using $n$ sums”, “approximate area by the shaded rectangles”, trapezoidal rule, using strips, approximate value of the definite integral.Your move Rectangles, strips or a table of values next to “estimate” or “approximate” tell you to add up shapes instead of integrating.
Area between and below curves	“Area between $y=\ldots$ and the $x$-axis from $x=a$ to $x=b$”, “total enclosed area”, “region bounded by …”, “state an integral expression representing the area”.Your move When the region is enclosed, bounded or between curves and nothing says estimate, integrate for the exact area.
Turning points and inflection	“Determine the stationary or turning point”, “the nature of the stationary point”, “point of inflection”, “show … has a point of inflection at $x=\ldots$”, the second derivative.Your move A named point is the target. Solve $f'=0$ for a turning point. For an inflection, solve $f''=0$ and check the sign changes.
Interpreting with the first and second derivative	Increasing or decreasing (the sign of $f'$), concave up or down (the sign of $f''$), “increasing most rapidly” (the maximum of $f'$), “identify the graph of the second derivative”.Your move Here you describe where the graph is increasing, decreasing or concave over an interval. One point on its own will not answer it.
Optimisation	“Determine the maximum or minimum [length, area, time]”, shortest, longest, greatest, least, minimise or maximise the total, “the value of $x$ that minimises …”.Your move Look for the words largest, smallest, maximum or minimum in a real context. Write the function, solve for where its derivative is $0$, then check it is the one you want.
Total change of a function	“Total change in the [quantity]”, distance travelled, change in displacement from $t=a$ to $t=b$, the total number or volume over the interval.Your move A rate is given and you need the total over time. Integrate the rate between the two times, using $\int v\,dt$ for displacement and $\int\|v\|\,dt$ for distance.