Skip to content
Longhand

Before the midterm

Dividing by a variable that could be zero

The habit: Factor instead, or handle the case where it vanishes on its own, so that no candidate is lost.

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

Practise against it: 3 questions

What it looks like

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

Challenging3 marks · 5 min

The join of the function below sits at x=cx = c, and the constant cc also appears in the first piece. Find every value of cc that makes ff continuous everywhere.

f(x)={cx2,x≤c9x,x>cf(x) = \begin{cases} cx^{2}, & x \leq c \\[4pt] 9x, & x > c \end{cases}
With the slip
c=3 only\displaystyle c = 3 \text{ only}
Worked through
c=−3, c=0 or c=3\displaystyle c = -3,\ c = 0 \text{ or } c = 3

Two roots lost: dividing c³ = 9c by c assumes c ≠ 0, and c² = 9 has a negative root as well as a positive one.

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 the interval 0<x<2π0 < x < 2\pi, where is f(x)=2sin⁡x+cos⁡(2x)f(x) = 2\sin x + \cos(2x) increasing?

    Curve sketchingStandard

  2. 02
    A student uses Lagrange multipliers to find the maximum value of the function below on the circle x2+y2=9x^{2} + y^{2} = 9, as shown. Find the first line that is wrong. f(x,y)=x2+3y2\displaystyle f(x, y) = x^{2} + 3y^{2}

    Optimization in two variables and Lagrange multipliersStandard

  3. 03
    Find every xx in the interval (0,2π)(0, 2\pi) at which the graph of the function below has a horizontal tangent line, and enter the sum of those values. f(x)=cos⁡x (1+sin⁡x)\displaystyle f(x) = \cos x\,(1 + \sin x)

    Exponential and trigonometric functionsChallenging

Start the set