Before the final

Treating a changing rate as constant

The habit: When the slope depends on the amount or the time, it has to be found again at each step, or the exact solution used; one slope held throughout is a straight line.

10 practice questions are built to catch it, in 3 topics.

See it fail in a figure: Hold one slope, in the slope fields and Euler’s method lesson

Practise against it: 6 questions

What it looks like

A question from exponential growth and decay, with the answer this slip gives beside the right one.

Introductory1 mark · 2 min

Which of the following functions satisfies the differential equation $\dfrac{dy}{dt} = 3y$ ?

With the slip: $\displaystyle y = 3t + 5$
Worked through: $\displaystyle y = 5e^{3t}$

3t + 5 climbs by 3 for every unit of t, whatever y is. dy/dt = 3y asks for growth three times the current amount, which speeds up as y grows.

See the worked solution

Where it comes up

Exponential growth and decay6 questions
Slope fields and Euler’s method3 questions
Antiderivatives1 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 quantity $y(t)$ satisfies $y'(t) = 3y(t)$ for all $t$ , and $y(0) = 4$ . Find $y(1)$ .
Exponential growth and decayIntroductory
02
The slope field of $\dfrac{dy}{dx} = (x - 1)(y + 1)$ is shown, with the point $(0, 1)$ marked. On which interval is the solution through $(0, 1)$ increasing?
Slope fields and Euler’s methodIntroductory
03
A radioactive isotope decays so that 30 % of any sample is gone after 10 days. How long does it take for a 200 g sample to decay to 50 g?
Exponential growth and decayStandard
04
A yeast culture grows at a rate proportional to its size, and increases by 50 % every 4 hours. What is its doubling time?
Exponential growth and decayStandard
05
A savings account pays 4 % per year, compounded continuously, and its owner deposits money continuously at $3000 per year. The balance $A(t)$ therefore satisfies $A'(t) = 0.04A + 3000$ . If the account opens with $20 000, what is the balance after 10 years?
Exponential growth and decayChallenging
06
A retiree has $100 000 in an account paying 5 % per year, compounded continuously, and withdraws money continuously at $6000 per year, so the balance satisfies $A'(t) = 0.05A - 6000$ . Solve for $A(t)$ , and find how many years it takes for the money to run out.
Exponential growth and decayCapstone

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