Before the midterm

Holding a changing quantity constant

The habit: Before differentiating, decide what moves with time and write it in terms of the variable you track; only the truly fixed dimensions stay as numbers.

8 practice questions are built to catch it, in 2 topics.

Practise against it: 6 questions

What it looks like

A question from related rates, with the answer this slip gives beside the right one.

Standard3 marks · 5 min

A bead slides along a wire bent into the curve $y = 2\sqrt{x}$ , with $x$ and $y$ in centimetres. Its $x$ -coordinate increases at 2 cm/s. At what rate is the bead’s distance from the origin changing as it passes the point $(4, 4)$ ?

Give a signed rate in centimetres per second.

With the slip: $\displaystyle \approx 1.414$
Worked through: $\displaystyle \frac{dD}{dt} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \approx 2.12 \text{ cm/s}$

√2 is (4 · 2)/(4√2), without the y y′ term. The bead stays on y = 2√x, so y rises at dy/dt = (1/√x) dx/dt = 1, adding 4 · 1 to the top.

See the worked solution

Where it comes up

Implicit differentiation and inverse trigonometric functions3 questions
Related rates5 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 glassmaker draws out a hot glass rod. The rod stays a solid cylinder of volume $100\pi$ cm³, with radius $r$ cm and length $L$ cm, while its radius shrinks at 0.1 cm per second. Two students find $\dfrac{dL}{dt}$ at the moment the radius is 2 cm, as shown. Which solution is right, and why?
Related ratesStandard
02
For the circle below, find $\dfrac{d^{2}y}{dx^{2}}$ at a point where $y > 0$ , in terms of $x$ and $y$ . $\displaystyle x^{2} + y^{2} = 25$
Implicit differentiation and inverse trigonometric functionsChallenging
03
The point $(-1, 2)$ lies on the curve below. Find $\dfrac{d^{2}y}{dx^{2}}$ at that point. $\displaystyle x^{3} + y^{3} = 7$
Implicit differentiation and inverse trigonometric functionsChallenging
04
Water drains from an inverted right circular cone at 8 cm³/s. The cone is 12 cm tall and 6 cm in radius at the top. At what rate is the water level changing when the water is 4 cm deep? $\displaystyle V = \frac{1}{3}\pi r^{2} h$
Related ratesChallenging
05
A balloon is released from the ground and rises straight up at 2 m/s. At the moment of release, a runner 30 m away runs straight toward the release point at 4 m/s. At what rate is the distance between the runner and the balloon changing 3 seconds after the release?
Related ratesChallenging
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

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