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Longhand

Before the midterm

Misjudging the number of real roots

The habit: The degree only bounds the count; the turning points and the sign of the function there decide it.

8 practice questions are built to catch it, in 5 topics.

Practise against it: 6 questions

What it looks like

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

Standard3 marks · 4 min

The join of the function below sits at x=cx = c, where cc is a constant. Find every value of cc that makes ff continuous everywhere, and enter the sum of those values.

f(x)={2x2−1,x≤c7x−4,x>cf(x) = \begin{cases} 2x^{2} - 1, & x \leq c \\[4pt] 7x - 4, & x > c \end{cases}
With the slip
3\displaystyle 3
Worked through
c=12 or c=3,sum 72\displaystyle c = \frac{1}{2} \text{ or } c = 3, \quad \text{sum } \frac{7}{2}

c = 3 is one root, but the condition is a quadratic in c, and it has two: find the other.

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
    How many steady states does this differential equation have? dydt=y4−4y2\displaystyle \frac{dy}{dt} = y^{4} - 4y^{2}

    Logistic growth and phase linesIntroductory

  2. 02
    How many real solutions does the equation below have? 3sin⁡(2x)=x\displaystyle 3\sin(2x) = x

    Functions, graphs and inversesStandard

  3. 03
    A function ff is continuous on [0,6][0, 6], and the table gives some of its values. What is the least number of zeros ff must have in [0,6][0, 6]?

    ContinuityStandard

  4. 04
    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.

    ContinuityChallenging

  5. 05
    How many real solutions does the equation ex=3xe^{x} = 3x have?

    Curve sketchingChallenging

  6. 06
    How many real roots does the function below have? p(x)=x4−4x+1\displaystyle p(x) = x^{4} - 4x + 1

    The Mean Value TheoremChallenging

Start the set