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Longhand

Before the midterm

Misjudging the domain

The habit: Zero under a root is allowed and zero in a denominator is not; check every candidate against the domain of the situation.

11 practice questions are built to catch it, in 7 topics.

See it fail in a figure: A short wall, in the optimization lesson

Practise against it: 6 questions

What it looks like

A question from continuity, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

On which set is g(x)=1x2−4g(x) = \dfrac{1}{\sqrt{x^{2} - 4}} continuous?

With the slip
(−∞,−2]∪[2,∞)\displaystyle (-\infty, -2] \cup [2, \infty)
Worked through
(−∞,−2)∪(2,∞)\displaystyle (-\infty, -2) \cup (2, \infty)

At x = ±2, x² − 4 = 0: the root of 0 is fine, but it sits in a denominator, so g(±2) would be 1/0, which is undefined.

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
    On which set is the function below continuous? f(x)=log⁡(4−x2)+1x−1\displaystyle f(x) = \log\bigl(4 - x^{2}\bigr) + \frac{1}{x - 1}

    ContinuityIntroductory

  2. 02
    On which set is the function below continuous? At an endpoint of its domain, count ff as continuous when its one-sided limit from inside the domain equals its value there. f(x)=5−xlog⁡x\displaystyle f(x) = \frac{\sqrt{5 - x}}{\log x}

    ContinuityIntroductory

  3. 03
    Let f(x)=1xf(x) = \dfrac{1}{x}, so that f(−1)=−1<0f(-1) = -1 < 0 and f(1)=1>0f(1) = 1 > 0. Does the Intermediate Value Theorem guarantee a number cc in (−1,1)(-1, 1) with f(c)=0f(c) = 0?

    ContinuityIntroductory

  4. 04
    The curve y=x3−3x2+5y = x^{3} - 3x^{2} + 5 has a tangent line parallel to y=9x+1y = 9x + 1 at exactly one point with x>0x > 0. Find the yy-intercept of that tangent line.

    Differentiation rulesStandard

  5. 05
    Two lamps stand 8 m apart on a straight path. At a distance of dd metres from a lamp, the brightness it gives is a power function of dd: 9d2\dfrac{9}{d^{2}} for the stronger lamp and 1d2\dfrac{1}{d^{2}} for the weaker one. At what distance from the stronger lamp, on the path between the two lamps, is the brightness from each lamp the same?

    Power functions and first sketchesChallenging

  6. 06
    Which statement about the vertical asymptotes of the graph of the function below is correct? g(x)=x2−4x+7−3\displaystyle g(x) = \frac{x^{2} - 4}{\sqrt{x + 7} - 3}

    LimitsChallenging

Start the set