Before the midterm

Misforming the difference quotient

The habit: f(a + h) minus f(a), over h, as h tends to 0: the same h in every place, and the subtraction in that order.

8 practice questions are built to catch it, in 1 topic.

Practise against it: 6 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

Which limit represents $g'(9)$ for $g(x) = \sqrt{x}$ ?

With the slip: $\displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{9 + h}$
Worked through: $\displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h}$

9 + h is the new input itself; from 9 to 9 + h the input changes by h.

See the worked solution

Where it comes up

The definition of the derivative8 questions

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 particle moves along a line with position $s(t) = t^{2} + 4t$ metres at time $t$ seconds. Find its average velocity over the interval from $t = 1$ to $t = 4$ .
The definition of the derivativeIntroductory
02
A student uses the definition of the derivative to find $f'(2)$ for the function below, as shown. Find the first line that is wrong. $\displaystyle f(x) = x^{2} + 3$
The definition of the derivativeStandard
03
A particle moves along a line, and its position at time $t$ seconds is $s(t) = 2t^{3} - 5t$ metres. Use the definition of the derivative to find its velocity $v(t)$ at time $t$ . $\displaystyle v(t) = \lim_{h \to 0} \frac{s(t + h) - s(t)}{h}$
The definition of the derivativeStandard
04
Let $f(x) = x\,|x|$ . Which statement about $f$ at $x = 0$ is correct?
The definition of the derivativeStandard
05
Let $f(x) = \dfrac{x}{x + 3}$ . Write the difference quotient $\dfrac{f(x + h) - f(x)}{h}$ as a single fraction in $x$ and $h$ , with the factor $h$ cancelled.
The definition of the derivativeStandard
06
A function $f$ is differentiable, with $f(3) = 5$ and $f'(3) = -2$ . Evaluate the limit below. $\displaystyle \lim_{h \to 0} \frac{f(3 + 2h) - f(3 - 3h)}{h}$
The definition of the derivativeChallenging