Before the midterm

Assuming a continuous function is differentiable

The habit: A derivative at a point forces the graph to be unbroken there, but not the reverse: |x| is unbroken at 0 and has a corner, so no slope.

5 practice questions are built to catch it, in 3 topics.

See it fail in a figure: Where there is no slope, in the definition of the derivative lesson

Practise against it: 4 questions

What it looks like

A question from the definition of the derivative, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

The function $f$ is piecewise linear: its graph is made of straight segments joining $(-2, 1)$ , $(1, 7)$ , $(3, 7)$ and $(5, -1)$ , in that order, as shown. Which statement about its derivative is correct?

With the slip: $f'(0) = 2$ , $f'(2) = 0$ , $f'(4) = -4$ , and $f'(1) = 1$
Worked through: $f'(0) = 2$ , $f'(2) = 0$ , $f'(4) = -4$ , and $f'(1)$ does not exist

The graph is unbroken at x = 1, but it turns a corner there: slope 2 just to the left, 0 just to the right. No single tangent fits a corner, and an average of the two slopes is not one.

See the worked solution

Where it comes up

Continuity2 questions
The definition of the derivative2 questions
Exponential and trigonometric functions1 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
A function $f$ is continuous at $x = 2$ , and $\lim_{x \to 2} f(x) = 5$ . Which of the following must be true?
ContinuityIntroductory
02
The graph of a function $f$ is shown. Of $x = -3$ , $x = -1$ , $x = 1$ and $x = 3$ , which set lists exactly the ones at which $f$ is not continuous?
ContinuityStandard
03
Let $f(x) = |x - 2|$ . Which statement about $f$ at $x = 2$ is correct?
The definition of the derivativeStandard
04
Let $f(x) = \sin|x|$ . Which statement about $f$ at $x = 0$ is true?
Exponential and trigonometric functionsChallenging

Start the set