L3 Differentiation
Question Bank
The value of a car, $V$ dollars, $t$ years after it was bought new is modelled by
$ V = 38\,000e^{-0.15t} + 4000 $
Find the rate at which the value of the car is changing 3 years after it was bought. Give your answer to the nearest dollar per year.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = \sqrt{x^2 + 9}$ at the point $(4, 5)$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = \tan(3x)$ at the point where $x = \dfrac{\pi}{12}$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The number of kākā in a fenced, predator-free sanctuary $t$ years after the fence was finished is modelled by
$ P = 150 + 40\sqrt{3t + 1} $
At what rate is the number of kākā increasing 8 years after the fence was finished?
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = x^2e^{-x}$ at the point where $x = 1$. Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
A loaf of bread is taken out of the oven. Its temperature, $T$ °C, $t$ minutes later is modelled by
$ T = 20 + 75e^{-0.04t} $
Find the rate at which the temperature of the loaf is changing 15 minutes after it comes out of the oven. Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = \operatorname{cosec}(2x)$ at the point where $x = \dfrac{\pi}{12}$. Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
The depth of water, $h$ metres, at the end of a wharf is modelled by
$ h = 3.5 + 1.2\cos(0.5t) $
where $t$ is the time in hours after midnight.
Is the depth of the water increasing or decreasing at 7 a.m.? Give a reason for your answer.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = \dfrac{x}{\ln x}$ at the point where $x = e^2$.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the exact $x$-coordinate of the point on the curve $y = e^{2x} - 6x$ where the tangent is parallel to the line $y = 4x + 1$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The temperature of the water in a rock pool, $T$ °C, $t$ hours after midnight, is modelled by
$ T = 17 - 3\cos\left(\frac{\pi(t - 2)}{12}\right) $
for $0 \le t \le 24$. Find the exact rate at which the water temperature is changing at 10 a.m.
You must use calculus and show any derivatives that you need to find when solving this problem.
The concentration of a medicine in a patient's blood, $C$ mg/L, $t$ hours after a dose is modelled by
$ C = 12te^{-0.4t} $
Is the concentration increasing or decreasing 3 hours after the dose? Give a reason for your answer.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = x\cot x$ at the point where $x = \dfrac{\pi}{4}$. Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the gradient of the tangent to the curve $y = \ln(\tan x)$, $0 < x < \dfrac{\pi}{2}$, at the point where $x = \dfrac{\pi}{6}$. Give your answer in exact form.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the $x$-coordinate(s) of the point(s) on the curve $y = 5\ln(x^2 + 4)$ where the gradient of the tangent is 2.
You must use calculus and show any derivatives that you need to find when solving this problem.
A curve has equation $y = \tan x - 2x$ for $0 < x < 2\pi$, $x \ne \dfrac{\pi}{2}, \dfrac{3\pi}{2}$.
Find the exact $x$-coordinates of all the points on the curve where the gradient of the tangent is 2.
You must use calculus and show any derivatives that you need to find when solving this problem.
A curve has equation $y = (x - k)^2e^{x}$, where $k$ is a constant.
The gradient of the tangent to the curve at the point where $x = 0$ is 8. Find the possible value(s) of $k$.
You must use calculus and show any derivatives that you need to find when solving this problem.
The curve $y = ae^{kx}$, where $a$ and $k$ are constants, passes through the point $(0, 8)$, and the gradient of the tangent at that point is $-2$.
Find the exact coordinates of the point on the curve where the gradient of the tangent is $-\dfrac{1}{2}$.
You must use calculus and show any derivatives that you need to find when solving this problem.
After a firing, a pottery kiln is switched off and left to cool. Its temperature, $T$ °C, $t$ hours after it is switched off is modelled by
$ T = 20 + 1180e^{-kt} $
where $k$ is a positive constant. Just after it is switched off, at $t = 0$, the kiln is cooling at 177 °C per hour.
How long after it is switched off is the kiln cooling at 20 °C per hour? Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.
Find the exact $x$-coordinates of all the points on the curve $y = e^{2x} - 5e^{x} + 4x$ where the tangent is parallel to the line $y = x + 7$.
You must use calculus and show any derivatives that you need to find when solving this problem.
At 10:00 a container ship leaves port, sailing due east at 18 km/h. At 11:00 a ferry leaves the same port, sailing due south at 30 km/h. Both ships keep the same speed and course.
How fast is the distance between the two ships increasing at 13:00? Give your answer correct to 3 significant figures.
You must use calculus and show any derivatives that you need to find when solving this problem.