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VICMath MethodsQuick questions
Unit 3
Quick questions on Tangents, stationary points and curve sketching: VCE Math Methods Unit 3
15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is tangent line?Show answer
At the point $(a, f(a))$ on $y = f(x)$, the tangent line has slope $f'(a)$ and passes through $(a, f(a))$. In point-slope form:
What is normal line?Show answer
The normal is perpendicular to the tangent at the same point, so its slope is $-\frac{1}{f'(a)}$ (provided $f'(a) \neq 0$):
What is standard pattern?Show answer
1. Compute $f(a)$ to get the y-coordinate of the point. 2. Compute $f'(a)$ to get the tangent slope.
What is first derivative test?Show answer
Check the sign of $f'(x)$ just before and just after the stationary point.
What is second derivative test?Show answer
Evaluate $f''(x)$ at the stationary point.
What is worked example?Show answer
Sketch $f(x) = x^3 - 6 x^2 + 9 x$ on $\mathbb{R}$.
What is intercepts?Show answer
y-intercept: $f(0) = 0$. x-intercepts: factor $x^3 - 6 x^2 + 9 x = x(x^2 - 6 x + 9) = x(x - 3)^2$. Zeros at $x = 0$ (single root) and $x = 3$ (double root).
What is stationary points?Show answer
$f'(x) = 3 x^2 - 12 x + 9 = 3(x^2 - 4 x + 3) = 3 (x - 1)(x - 3)$. Stationary at $x = 1$ and $x = 3$.
What is point of inflection?Show answer
$f''(x) = 0$ at $x = 2$. $f(2) = 8 - 24 + 18 = 2$. Inflection at $(2, 2)$.
What is end behaviour?Show answer
Cubic with positive leading coefficient: $f \to -\infty$ as $x \to -\infty$, $f \to +\infty$ as $x \to +\infty$.
What is forgetting to classify stationary points?Show answer
Setting $f'(x) = 0$ only finds candidates. You must justify whether each is a maximum, minimum or stationary point of inflection.
What is assuming $f'' = 0$ guarantees an inflection?Show answer
Concavity must actually change. For $f(x) = x^4$, $f''(0) = 0$ but no inflection there.
What is wrong slope for the normal?Show answer
The normal slope is $-\frac{1}{f'(a)}$, not $-f'(a)$ or $\frac{1}{f'(a)}$.
What is mixing up the y-coordinate?Show answer
When writing the equation $y - f(a) = f'(a)(x - a)$, $f(a)$ is the y-coordinate at the point, not the slope.
What is missing repeated roots when sketching?Show answer
A double root means the curve touches the x-axis without crossing. Drawing it as a crossing loses a mark.