Chapter 0: The basics

Power functions and first sketches

The course opens with the simplest functions there are, the powers of x, and one question about them: for a given x, which of two powers is bigger? Answering it for very small and very large x is enough to sketch any two-term polynomial, to locate the intersections of power functions, and to see why an exponential eventually beats every power. It is also the language the rest of the course uses for limits at 0 and at infinity.

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Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

Introductory1 mark · 2 min
Let $f(x) = 3x^{2} - x^{5}$ . For very large positive $x$ , which term dominates, and what is the sign of $f(x)$ ?
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Introductory1 mark · 2 min
Let $g(x) = x^{4} + 6x$ . Close to $x = 0$ , which term dominates, and what does the graph look like there?
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Introductory2 marks · 3 min
The curves $y = 2x^{3}$ and $y = 8x$ meet at three points. Find the largest $x$ -coordinate at which they meet.
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Standard2 marks · 3 min
Find the positive value of $x$ at which the curves below intersect.
$y = x^{4} \qquad \text{and} \qquad y = 5x^{2}$
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Standard2 marks · 4 min
Which description matches the graph of $y = x^{3} - x^{5}$ ?
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Standard3 marks · 4 min
The speed of a reaction depends on the concentration $x$ of a chemical according to the function below. As $x$ grows large, $y$ approaches a limiting value. Find the concentration $x > 0$ at which $y$ is exactly half of that limiting value.
$y = \frac{6x^{3}}{2 + x^{3}}$
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Standard2 marks · 3 min
For all sufficiently large $x$ , which ordering of the three functions $e^{x}$ , $x^{100}$ and $\ln x$ is correct, from largest to smallest?
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Challenging3 marks · 5 min
Find the value of $x > 0$ at which the curves below intersect.
$y = 4\sqrt{x} \qquad \text{and} \qquad y = x^{2}$
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Challenging3 marks · 6 min
Two fish in a school keep a preferred distance apart. When they are $x$ metres apart, an attractive force pulls them together and a repulsive force pushes them apart, with strengths given below. The preferred spacing is the distance at which the two forces balance exactly. Find it.
$F_{\text{attract}}(x) = \frac{3}{x^{2}}, \qquad F_{\text{repel}}(x) = \frac{48}{x^{6}}$
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Capstone5 marks · 9 min
Let $y = \rho x - x^{5}$ , where $\rho$ is a nonzero constant. Which statement about the graph is correct for both signs of $\rho$ ?
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