Before the midterm

Confusing the intercept with the value at the point

The habit: The y-intercept is the line’s height at x = 0; the height it shares with the curve is at x = a, where the two touch.

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

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.

Standard3 marks · 5 min

Use the definition of the derivative to find the slope of the tangent line to $y = \dfrac{1}{x}$ at $x = 2$ . Then find the $y$ -intercept of that tangent line, and enter it.

Rules such as the power rule earn no marks on this question.

With the slip: $\displaystyle 0.5$
Worked through: $\displaystyle y = -\frac{1}{4}x + 1;\quad \text{intercept } 1$

1/2 is the height of the curve at x = 2, where the line touches it. The intercept is the line’s height two units to the left, at x = 0, and the slope says how much that changes.

See the worked solution

Where it comes up

The definition of the derivative2 questions
Differentiation rules2 questions
Implicit differentiation and inverse trigonometric functions2 questions
Linear approximation1 question
Newton’s method1 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
Find an equation of the tangent line to $y = 2x^{3} - x$ at the point where $x = 2$ .
Differentiation rulesIntroductory
02
The linear approximation of a function $f$ about $x = 2$ is $L(x) = 4x - 7$ . What are $f(2)$ and $f'(2)$ ?
Linear approximationIntroductory
03
Let $f(x) = \sqrt{10 - 3x}$ . Use the definition of the derivative to find $f'(2)$ , and then find an equation of the tangent line to $y = f(x)$ at $x = 2$ .
The definition of the derivativeStandard
04
The curve $y = x^{3} - 3x^{2} + 5$ has a tangent line parallel to $y = 9x + 1$ at exactly one point with $x > 0$ . Find the $y$ -intercept of that tangent line.
Differentiation rulesStandard
05
Find the $x$ -intercept of the line tangent to the curve $y = x\arctan x$ at the point where $x = 1$ .
Implicit differentiation and inverse trigonometric functionsStandard
06
The graph of $y = \arctan x$ has exactly two tangent lines parallel to the line $y = \tfrac{1}{2}x$ . Find the $y$ -intercept of the one that touches the graph at a point with $x < 0$ .
Implicit differentiation and inverse trigonometric functionsStandard

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