How to Tackle the E Questions

L2MAEL

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Throughout the past 12 years, NZQA consistently have given us the same content-based questions.

These are the most common types of excellence questions, but is not limited to:

• Maximizing/minimizing
• Kinematics
• Increasing/decreasing functions
• Optimization (graphic & geometric)
• Differentiation with 1 or more unknown variables (constant)
• Rate of change
• Antidifferentiation with 1 or more unknown variables (constant)
• Sketching

Use the given information. The most common information given is “has a stationary point at $(x,y)$“. Although it is 1 sentence, there are two information given there. One is that "$f^{\prime }(x)=0$", because it’s at station, and the other is “curve passes through $(x,y)$“.

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1. Maximizing & Minimizing: Maximizing and minimizing is one of the most crucial, and most basic excellence question that could come up. The key here is to always prove that your given answer is the maximum/minimum by either:

• second derivative test ($f^{\prime \prime }(x),\frac{d^{2}y}{d{x}^2}$)
• inspection of $y$-coordinate
• shape of graph
• checking gradient of tangent at points around the stationary point

If you don’t know how to maximize/minimize something, you probably haven’t paid attention in class, since this involves the most basic knowledge of differentiation.

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2. Kinematics: Kinematics involves $a(t),v(t),s(t)$ — acceleration, velocity, and displacement respectively. First thing you should do when you see a kinematics question is that you should understand what you’re looking for.

e.g. If you’re looking for the displacement (or distance), and you only have $a(t)$, then you know that you have to antidifferentiate it twice.

When antidifferentiated, remember the constant of integration. the $+C$ at the end.

The integration constant is usually calculated with additional information given in the question. The most common one is that ”??? starts from rest”, which means that $v(0)=0$, which also means that when time is zero, velocity is zero. This also means that you start timing at zero, which implies that when you start the timer $t=0$, its distance is also zero $⟹s(t)=0$!

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3. Increasing & Decreasing Functions: The FIRST thing you should do when solving these questions is to DRAW the graph. Doesn’t have to be neat, doesn’t have to have an axis. Just need the shape of the graph. Look at the highest power in the function, and the sign of it.

For example,

$$f(x)=x^3+ax+bx+c$$

Since its highest degree term is $x^3$, its going to be a cubic, and because the sign of the highest degree is a positive, I know that this is a positive cubic function.

You should be familiar with drawing quadratics all the way up to quartics; if not, practice.

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4. Optimization: The FIRST thing you should do is write down the formula of what you’re maximizing.

If you’re maximizing the area of a triangle, write down the formula for the area of the triangle:

$$A=\frac{1}{2}\cdot b\cdot h$$

Likewise, if you’re maximizing the volume of a cuboid, write down the formula for the volume of the cuboid:

$$V=l\cdot w\cdot h$$

Then, you should try and make the formula in terms of only 1 variable. This is done usually through clues in the question.

If you get an optimization question that involves the axis, you should know that any point on the graph can be expressed as (x,y).

Once in terms of 1 variable, differentiate it, set it to zero, then prove that it’s maximum/minimum.

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5. Differentiation with 1 or more variables (constant): This often involves the use of simultaneous equations (when there are $2+$ variables). For 2 variables, you will need 2 equations. First equation is the function itself, whereas the second equation is the derivative.

The most common information given is ”$(x,y)$ is a stationary point”. This actually gives you two informations — one is that $f(x)=y$, and $f^{\prime }(x)=0$. Using these two information, you will be able to get both the original equation and the derivative equation in terms of only the unknown variables, and you’ll be able to solve the system.

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6. Rate of Change: Rate of change is nothing different to normal differentiation. For example when you get an equation for a volume:

$$V=3x^4+2x^3+x^2+1$$

The rate at which the volume changes is

$$\frac{dV}{dt}=12x^3+6x^2+2x$$

Usually, you are given the rate in the question, and you’re asked to manipulate that to find the answer.

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7. Antidifferentiation with 1 or more variables (constant): This also involves the use of simultaneous equations. Check number 5 for more info.

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8. Sketching: Although sketching the graph as an excellence question is rare, it can still come out. In fact, sketching has always been in the question. Either as an achieved or a merit question, but sometimes it can be excellence.

The excellence-level sketching is no different to the other sketching questions, except you have to find the equation yourself.

A tip here is that you should show all your intercepts, vertex, turning points, and other features of the graph CLEARLY.

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