L2 Calculus
Question Bank
The cost, $\$C$, to produce $x$ items for a company is modelled by $C(x) = 0.1x^3 - 3x^2 + 50x$. The marginal cost is the rate of change of the cost, $C'(x)$. Find the marginal cost when 20 items are produced.
The area, $A$, of a square changes as the length $s$ of its sides changes.
$A = s^2$
Find the rate of change of the area, with respect to the side length, when the side length is 8 cm.
The surface area, $A$, of a sphere changes as the radius $r$ of the sphere changes.
$A = 4\pi r^2$
Find the rate of change of the surface area, with respect to the radius, when the radius is 3 cm.
A car's distance from its starting point is given by $s(t) = t^3 - 6t^2 + 9t$, where $s$ is in metres and $t$ is in seconds.
At which time is the car travelling faster: at $t = 0.5$ seconds or at $t = 3$ seconds?
A cylindrical water tank has a fixed height of 10 metres. The volume, $V$, of water in the tank changes as the radius $r$ changes.
$V = 10\pi r^2$
Find the rate of change of the volume, with respect to the radius, when the radius is 2 metres.
The population of a town is modelled by $P(t)$ $= 2t^3 - 15t^2 + 36t + 5000$, where $P$ is the population and $t$ is the time in years after 2020.
Is the population increasing faster at $t = 1$ year or at $t = 4$ years?
At the opening of a new restaurant, 50 customers were present. After $t$ hours from opening, the number of customers, $C$, can be modelled by
$C(t) = 50 + 30t + 5t^2$
How many hours will it take for the rate of change of the number of customers to be 50 customers per hour?
A company's profit is modelled by $P(x)$ $= -2x^3 + 27x^2 + 60x + 100$, where $P$ is the profit in thousands of dollars and $x$ is the number of years since the company started.
Is the profit growing faster at $x = 2$ years or at $x = 6$ years?
The profit, $\$P$, from selling $x$ items is modelled by $P(x) = 2x^2 + 50x - 300$. Find how many items are being sold when the rate of change of profit is $\$70$ per item.
A circular oil slick is expanding. Find the rate of change of its area with respect to its radius, $r$, at the instant when the area is $16\pi$ m².
A square puddle is evaporating. Its side length, $s$, starts at 25 cm and decreases at a constant rate of 0.5 cm/min. This means the side length at time $t$ minutes is given by the formula $s = 25 - 0.5t$.
Find the rate at which the area, $A$, is decreasing at the instant when the side length is 20 cm.
The total number of views, $V$ thousand, of a video $t$ days after it is posted online is modelled by
$V = -t^3 + 12t^2 + 5t$
for $0 \le t \le 8$.
Use calculus to find when the number of views is increasing fastest, and the rate of increase at that time.
An ice cube melts so that its side length, $x$ cm, after $t$ minutes is given by $x = 6 - 0.2t$.
The volume of the cube is $V = x^3$.
Find the rate at which the volume of the ice cube is decreasing at the instant when its side length is 4 cm.
Air is being pumped into a spherical balloon. The radius, $r$, is increasing at a constant rate of 0.5 cm/s and has an initial radius of 1 cm. This means the radius at time $t$ seconds is given by the formula $r = 0.5t + 1$.
Find the rate at which the volume, $V$, is increasing at the instant when the radius is 4 cm. (Note: $V = \frac{4}{3}\pi r^3$)
The height, $h$ cm, of a sunflower $t$ days after its seed is planted is modelled by
$h = -0.002t^3 + 0.18t^2$
for $0 \le t \le 60$.
Use calculus to find the height of the sunflower at the time when it is growing fastest. Justify that this is when the growth rate is greatest.