Before Quiz 2

Using the wrong derivative for the question

The habit: f′ says where the graph rises and falls, f″ how it bends and where it inflects, so match the order to what is asked.

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

Practise against it: 6 questions

What it looks like

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

Introductory1 mark · 2 min

Find the $x$ -coordinate of the inflection point of $f(x) = x^{3} - 6x^{2} + x$ .

With the slip: $\displaystyle \approx 3.915$
Worked through: $\displaystyle x = 2$

3.915 is a root of f′(x) = 3x² − 12x + 1, where f has a local minimum. An inflection point is where f″ changes sign.

See the worked solution

Where it comes up

Power functions and first sketches1 question
The definition of the derivative2 questions
Differentiation rules2 questions
Exponential and trigonometric functions2 questions
Implicit differentiation and inverse trigonometric functions3 questions
Curve sketching11 questions
Optimization1 question
Linear approximation1 question
Taylor polynomials1 question
Newton’s method3 questions
Slope fields and Euler’s method1 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 $g(x) = x^{4} + 6x$ . Close to $x = 0$ , which term dominates, and what does the graph look like there?
Power functions and first sketchesIntroductory
02
A roastery’s cost of roasting $q$ kilograms of coffee in a day is $C(q)$ dollars, and $C'(50) = 12$ . Which statement is the best reading of $C'(50) = 12$ ?
The definition of the derivativeIntroductory
03
Let $f(x) = \sin x$ . Find $f^{(50)}(x)$ , the 50th derivative of $f$ .
Exponential and trigonometric functionsIntroductory
04
A particle moves along a line, and its position at time $t$ seconds is $s(t) = t^{3} - 6t^{2} + 32$ metres. Which describes its motion at $t = 3$ ?
Differentiation rulesStandard
05
The graph of a function $f$ is shown. For $-3 \leq x \leq 0$ it is the arch $y = 1 - (x + 1)^{2}$ ; then a straight segment climbs from $(0, 0)$ to $(2, 2)$ , and the graph stays level at height $2$ after that. Which graph is the graph of $f'$ ?
The definition of the derivativeChallenging
06
The table gives values at $x = 2$ of two twice-differentiable functions $f$ and $g$ and of their first and second derivatives. Let $h(x) = f(x)\,g(x)$ . Find $h''(2)$ .
Differentiation rulesChallenging