Choosing and justifying a method

Covers 93202

Why the choice matters more than the computation

At Level 3 the question usually tells you the method ("use integration by parts"). At Scholarship it does not.
Marks are awarded for selecting a sensible approach and saying why it fits — that is the "analysis and critical thinking" in the standard.
Two students can reach the same answer and score very differently, purely on whether the reasoning is visible.

A routine that works

Write what you are given and what you want. One line each. This alone often reveals the route.
Name the structure. Is it a rate of change? An accumulation? A maximum? A relationship between two changing quantities?
Choose the tool and justify it in one sentence. "Both quantities change with time, so differentiate implicitly with respect to $t$ ."
Check the conditions. Is the function continuous? Differentiable on the interval? Is the denominator ever zero?
Do the algebra, showing intermediate results.
Interpret — what does the number mean, and is it reasonable?

Signals that point to a tool

"Rate", "per second", "how fast" → differentiate; if two quantities are linked, differentiate the relationship implicitly.
"Total", "accumulated", "area", "volume" → integrate, and be careful about the limits.
"Greatest", "least", "optimum" → differentiate, solve $f'(x)=0$ , then justify it is a maximum or minimum.
A parameter such as $k$ in the question → the answer is expected in terms of $k$ ; do not substitute a number.
"Show that..." → the answer is given, so all the credit is in the reasoning. Never work backwards from it.

Worked Example

A curve is given by $x^2 + xy + y^2 = 7$ .

Find the gradient of the curve at the point $(1, 2)$ , and state what the sign of your answer tells you.

Answer:

We cannot easily write $y$ as a function of $x$ here, so we differentiate implicitly with respect to $x$ — this is the tool that fits a relationship where both variables are entangled.

Differentiating each term:

$2x + \left(y + x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0$

The middle term needs the product rule, because $xy$ is a product of two functions of $x$ .

Collecting the $\dfrac{dy}{dx}$ terms:

$\frac{dy}{dx}(x + 2y) = -(2x + y)$

$\frac{dy}{dx} = -\frac{2x + y}{x + 2y}$

Substituting $x = 1$ , $y = 2$ :

$\frac{dy}{dx} = -\frac{2(1) + 2}{1 + 2(2)} = -\frac{4}{5}$

The gradient is $-\dfrac{4}{5}$ .

Because the gradient is negative, $y$ is decreasing with respect to $x$ at that point — moving along the curve in the direction of increasing $x$ , the curve falls.

Test yourself

Choosing and justifying a method

Covers 93202

Why the choice matters more than the computation

At Level 3 the question usually tells you the method ("use integration by parts"). At Scholarship it does not.
Marks are awarded for selecting a sensible approach and saying why it fits — that is the "analysis and critical thinking" in the standard.
Two students can reach the same answer and score very differently, purely on whether the reasoning is visible.

A routine that works

Write what you are given and what you want. One line each. This alone often reveals the route.
Name the structure. Is it a rate of change? An accumulation? A maximum? A relationship between two changing quantities?
Choose the tool and justify it in one sentence. "Both quantities change with time, so differentiate implicitly with respect to $t$ ."
Check the conditions. Is the function continuous? Differentiable on the interval? Is the denominator ever zero?
Do the algebra, showing intermediate results.
Interpret — what does the number mean, and is it reasonable?

Signals that point to a tool

"Rate", "per second", "how fast" → differentiate; if two quantities are linked, differentiate the relationship implicitly.
"Total", "accumulated", "area", "volume" → integrate, and be careful about the limits.
"Greatest", "least", "optimum" → differentiate, solve $f'(x)=0$ , then justify it is a maximum or minimum.
A parameter such as $k$ in the question → the answer is expected in terms of $k$ ; do not substitute a number.
"Show that..." → the answer is given, so all the credit is in the reasoning. Never work backwards from it.

Worked Example

A curve is given by $x^2 + xy + y^2 = 7$ .

Find the gradient of the curve at the point $(1, 2)$ , and state what the sign of your answer tells you.

Answer:

We cannot easily write $y$ as a function of $x$ here, so we differentiate implicitly with respect to $x$ — this is the tool that fits a relationship where both variables are entangled.

Differentiating each term:

$2x + \left(y + x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0$

The middle term needs the product rule, because $xy$ is a product of two functions of $x$ .

Collecting the $\dfrac{dy}{dx}$ terms:

$\frac{dy}{dx}(x + 2y) = -(2x + y)$

$\frac{dy}{dx} = -\frac{2x + y}{x + 2y}$

Substituting $x = 1$ , $y = 2$ :

$\frac{dy}{dx} = -\frac{2(1) + 2}{1 + 2(2)} = -\frac{4}{5}$

The gradient is $-\dfrac{4}{5}$ .

Because the gradient is negative, $y$ is decreasing with respect to $x$ at that point — moving along the curve in the direction of increasing $x$ , the curve falls.