Before the final

Evaluating the derivative at the wrong point

The habit: The derivative is taken at the base point, the point of tangency on the curve; find that point before substituting.

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

Practise against it: 6 questions

What it looks like

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

Introductory1 mark · 2 min

Use a linear approximation of $f(x) = \sqrt{x}$ at $x = 25$ to estimate $\sqrt{26}$ .

With the slip: $\displaystyle \approx 5.098$
Worked through: $\displaystyle \sqrt{26} \approx 5.1$

1/(2√26) is the slope at 26, which needs the very root being estimated. The line is built at 25, from f(25) and f′(25).

See the worked solution

Where it comes up

Differentiation rules1 question
Exponential and trigonometric functions1 question
Implicit differentiation and inverse trigonometric functions1 question
L’Hôpital’s rule1 question
Linear approximation4 questions
Taylor polynomials2 questions
Newton’s method2 questions
Slope fields and Euler’s method2 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
Find the linear approximation $L(x)$ of $f(x) = \log x$ at $x = e$ , where $\log$ is the natural logarithm.
Linear approximationIntroductory
02
The functions $f$ and $g$ have continuous derivatives everywhere, with $f(2) = 0$ , $f'(2) = 4$ , $g(2) = 3$ , $g'(2) = 1$ , $g(6) = 0$ and $g'(6) = 5$ . Evaluate the limit. $\displaystyle \lim_{x \to 2} \frac{f(x)}{g(3x)}$
L’Hôpital’s ruleStandard
03
Use a linear approximation to estimate $\arctan(0.9)$ , using the fact that $\arctan 1 = \dfrac{\pi}{4}$ .
Linear approximationStandard
04
Two distinct lines pass through the point $(1, -6)$ and are tangent to the parabola $y = 2x^{2}$ . Find the slope of the steeper one.
Differentiation rulesChallenging
05
Find the constants $a$ and $b$ for which the function below is differentiable everywhere, and enter $a + b$ .
Exponential and trigonometric functionsChallenging
06
The equation below defines $y$ as a differentiable function of $x$ for every real $x$ . Find $\dfrac{d^{2}y}{dx^{2}}$ at $x = 1 + \dfrac{\pi}{4}$ . $\displaystyle y + \arctan y = x$
Implicit differentiation and inverse trigonometric functionsCapstone