Before Quiz 2

Reading where f is positive as where it increases

The habit: The sign of f′, not the sign of f, says whether the graph rises or falls.

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

Practise against it: 5 questions

What it looks like

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

Introductory2 marks · 3 min

Find the largest open interval on which $f(x) = \dfrac{e^{x}}{x + 2}$ is increasing.

With the slip: $\displaystyle (-2, \infty)$
Worked through: $\displaystyle (-1, \infty)$

f is positive on (−2, ∞), but positive is not rising: f′ = e^x(x + 1)/(x + 2)², so check where that is positive.

See the worked solution

Where it comes up

Differentiation rules1 question
Exponential and trigonometric functions1 question
Curve sketching2 questions
Slope fields and Euler’s method1 question
Functions of two variables and partial derivatives1 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
The graph of a function $f$ is shown with four labelled points. At which point are $f'(x)$ and $f''(x)$ both negative?
Curve sketchingIntroductory
02
The slope field of $\dfrac{dy}{dx} = (x - 1)(y + 1)$ is shown, with the point $(0, 1)$ marked. On which interval is the solution through $(0, 1)$ increasing?
Slope fields and Euler’s methodIntroductory
03
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
04
What is the largest open interval on which $f(x) = xe^{x}$ is increasing?
Exponential and trigonometric functionsStandard
05
The surface $z = f(x, y) = 9 - x^{2} + xy - y^{2}$ is sliced by the vertical plane $y = 1$ . The slice is the curve $z = f(x, 1)$ , drawn below with $x$ across and $z$ up. Which statement about the slice at the marked point, where $x = 2$ , is correct?
Functions of two variables and partial derivativesStandard