Before the final

Misusing the step in a linear approximation

The habit: The change in output is the slope times the change in input from the base point: Δy ≈ f′(a) Δx.

8 practice questions are built to catch it, in 4 topics.

See it fail in a figure: The step from the base point, in the linear approximation lesson

Practise against it: 6 questions

What it looks like

A question from linear approximation, with the answer this slip gives beside the right one.

Introductory2 marks · 2 min

Find the linear approximation $L(x)$ of $f(x) = \log x$ at $x = e$ , where $\log$ is the natural logarithm.

With the slip: $\displaystyle 1 + \frac{x}{e}$
Worked through: $\displaystyle L(x) = 1 + \frac{1}{e}(x - e) = \frac{x}{e}$

That line multiplies the slope 1/e by x, so at x = e it gives 2, not log e = 1. The slope multiplies x − e, the distance from e.

See the worked solution

Where it comes up

Linear approximation4 questions
Taylor polynomials2 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
A loaf of bread comes out of the oven at 95 °C into a kitchen at 25 °C. Its temperature $T(t)$ , in °C after $t$ minutes, satisfies $\dfrac{dT}{dt} = 0.05\,(25 - T)$ . Use one step of Euler’s method with $\Delta t = 2$ to estimate the temperature after 2 minutes.
Slope fields and Euler’s methodIntroductory
02
Use a linear approximation to estimate $(3.002)^{4}$ .
Linear approximationStandard
03
A student uses the linear approximation of the function below at $x = 8$ to estimate $\sqrt[3]{8.3}$ , as shown. Find the first line that is wrong. $\displaystyle f(x) = \sqrt[3]{x}$
Linear approximationStandard
04
Find the second-order Taylor polynomial $T_{2}(x)$ of $f(x) = \sin x$ about $x = \dfrac{\pi}{6}$ .
Taylor polynomialsStandard
05
Find the third-order Taylor polynomial $T_{3}(x)$ of $f(x) = \dfrac{1}{x}$ about $x = 2$ .
Taylor polynomialsStandard
06
The concentration of a drug in the blood, $t$ hours after a dose, is modelled by $C(t) = 8t\,e^{-t/4}$ mg/L. Find the linear approximation $L(t)$ of $C$ at $t = 2$ .
Linear approximationChallenging