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Longhand

Before the midterm

Misreading a negative exponent

The habit: A minus sign upstairs means a reciprocal: (1/4)−1/2 = 41/2 = 2.

4 practice questions are built to catch it, in 4 topics.

Practise against it: 3 questions

What it looks like

A question from power functions and first sketches, with the answer this slip gives beside the right one.

Introductory1 mark · 2 min

The graph of one of these power functions is shown, with the points (−1,−1)(-1, -1) and (1,1)(1, 1) marked. Which function is it?

A curve drawn for x from −3 to 3, passing through the marked points (−1, −1) and (1, 1) and through the origin. It is very steep, almost vertical, as it passes through the origin, and flattens out away from it: at x = 3 it has only reached a height of about 1.4, and at x = −3 a depth of about −1.4.-3-2-1123-2-112
With the slip
y=x−2\displaystyle y = x^{-2}
Worked through
y=x1/3\displaystyle y = x^{1/3}

x−2 = 1/x² is not defined at 0 and is positive everywhere else. This graph passes through the origin and is negative for x < 0.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Let f(x)=x2x−6x3f(x) = x^{2}\sqrt{x} - \dfrac{6}{\sqrt[3]{x}} for x>0x > 0. Find f′(x)f'(x).

    Differentiation rulesIntroductory

  2. 02
    Evaluate the limit. lim⁡x→∞3x2−log⁡xx2+e−x\displaystyle \lim_{x \to \infty} \frac{3x^{2} - \log x}{x^{2} + e^{-x}}

    L’Hôpital’s ruleIntroductory

  3. 03
    Use the definition of the derivative to find f′(x)f'(x) for the function below. f(x)=1x2\displaystyle f(x) = \frac{1}{x^{2}}

    The definition of the derivativeStandard

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