Before the midterm

Not undoing the exponent

The habit: After isolating a power of x, take the matching root, and substitute the answer back to check.

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

What it looks like

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

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}

With the slip: $\displaystyle 5$
Worked through: $\displaystyle x = \sqrt{5} \approx 2.236$

After cancelling x², what remains is x² = 5: the square still has to be undone. At x = 5 the curves do not meet: x⁴ = 625 but 5x² = 125.

See the worked solution

Where it comes up

Power functions and first sketches3 questions
Functions, graphs and inverses1 question
Implicit differentiation and inverse trigonometric functions1 question
Optimization1 question
Newton’s method1 question
Functions of two variables and partial derivatives1 question
The Mean Value Theorem1 question

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.

01
Let $f(x) = 3e^{2x} - 1$ . Find $f^{-1}(5)$ .
Functions, graphs and inversesStandard
02
Find the exact value of the expression below. $\displaystyle \cos\bigl(\arctan(-2)\bigr)$
Implicit differentiation and inverse trigonometric functionsStandard
03
Find the value of $x > 0$ at which the curves below intersect. $\displaystyle y = 4\sqrt{x} \qquad \text{and} \qquad y = x^{2}$
Power functions and first sketchesChallenging
04
Two lamps stand 8 m apart on a straight path. At a distance of $d$ metres from a lamp, the brightness it gives is a power function of $d$ : $\dfrac{9}{d^{2}}$ for the stronger lamp and $\dfrac{1}{d^{2}}$ 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
05
A bird feeds in patches of berries. After $t$ minutes in a patch it has gained $f(t)$ units of energy, given below, so the patch gives up food more and more slowly. Flying to the next patch takes 6 minutes, with no food on the way. How long should the bird stay in each patch to make its long-run rate of energy gain, energy per minute of feeding and flying together, as large as possible? $\displaystyle f(t) = \frac{60t}{t + 3}$
OptimizationChallenging
06
Newton’s method for a root of $f(x) = x^{3} - 7x$ is started at some $x_{0} > 0$ . For exactly one such $x_{0}$ , the first step lands on $x_{1} = -x_{0}$ , and from then on the estimates alternate $x_{0}, -x_{0}, x_{0}, \ldots$ for ever without approaching a root. Find that $x_{0}$ .
Newton’s methodChallenging

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