How do you show a quantity V is decreasing in a differential equation?

\frac{dV}{dt} is negative, e.g. \frac{dV}{dt}=-kV with k0.

What is the connected rates of change formula for V, r and t?

\frac{dV}{dt}=\frac{dV}{dr}\times\frac{dr}{dt}

What does a rate of change of 0 mean in a model?

The quantity is constant: an equilibrium.

Should you substitute a value of r before or after differentiating?

After. Differentiate in terms of r first.

If \frac{dV}{dt}=-2, what does this tell you?

The volume is decreasing at a rate of 2 units per unit time.

Forming differential equations and connected rates of changeEdexcel International A Level Maths: Flashcards

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How do you write 'the rate of change of $y$ is proportional to $y$'?

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How do you write 'the rate of change of $y$ is proportional to $y$ '?: $\frac{dy}{dt}=ky$
How do you show a quantity $V$ is decreasing in a differential equation?: $\frac{dV}{dt}$ is negative, e.g. $\frac{dV}{dt}=-kV$ with $k>0$ .
Differential equation for 'rate of increase of $N$ is proportional to $\sqrt N$ '?: $\frac{dN}{dt}=k\sqrt N$
Differential equation for 'rate of change of $y$ is inversely proportional to $y$ '?: $\frac{dy}{dt}=\frac ky$
What is the connected rates of change formula for $V$ , $r$ and $t$ ?: $\frac{dV}{dt}=\frac{dV}{dr}\times\frac{dr}{dt}$
A sphere has $V=\frac43\pi r^3$ . Find $\frac{dV}{dr}$ .: $4\pi r^2$
A sphere has $S=4\pi r^2$ . Find $\frac{dS}{dr}$ .: $8\pi r$
How do you form a differential equation with inflow and outflow?: $\frac{dV}{dt}=\text{rate in}-\text{rate out}$
What does a rate of change of $0$ mean in a model?: The quantity is constant: an equilibrium.
A cube of edge $x$ has $V=x^3$ . Find $\frac{dV}{dt}$ in terms of $x$ and $\frac{dx}{dt}$ .: $3x^2\frac{dx}{dt}$
Should you substitute a value of $r$ before or after differentiating?: After. Differentiate in terms of $r$ first.
If $\frac{dV}{dt}=-2$ , what does this tell you?: The volume is decreasing at a rate of $2$ units per unit time.

Exam questions on Forming differential equations and connected rates of change

At time $t$ minutes a leaking tank holds $V$ m $^3$ of water. Water leaks out at a rate proportional to the volume $V$ of water remaining in the tank.
Find the rate at which the volume is changing when $V=80$ .2 marks
A spherical balloon is inflated so that its radius $r$ cm increases at a constant rate of $0.5$ cm s $^{-1}$ . The volume of a sphere is $V=\frac43\pi r^3$ and its surface area is $S=4\pi r^2$ .
Find the rate at which the surface area is increasing when $r=6$ .2 marks
Liquid is poured at a constant rate of $20$ cm $^3$ s $^{-1}$ into an inverted cone with its axis vertical. At time $t$ seconds the depth of liquid is $h$ cm and its volume is $V=\frac13\pi h^3$ cm $^3$ .
Find the rate at which the depth is increasing when $h=5$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).