L2 Calculus
Question Bank
Find the antiderivative of $f(x) = 6x^2 - 4x + 5$.
The gradient function for a curve is $f'(x) = 3x^2 - 5$. The curve passes through the point $(2, 0)$. Find the equation of the function $f(x)$.
The gradient function for a curve is $\frac{dy}{dx} = 8x^3 - 3$. The curve passes through the point $(1, 4)$. Find the equation for y.
Water flows into a tank at a rate of
$\frac{dV}{dt} = 3t^2 + 2$
litres per minute, where $t$ is the time in minutes after a tap is turned on. When the tap is turned on, the tank already holds 10 litres.
How much water is in the tank after 4 minutes?
The graphs of a function $f(x)$ and its derivative $f'(x)$ are shown. Use information from the graphs to find the equation of the function $f(x)$.
A curve $y = f(x)$ passes through the point $(2, 10)$ and has a gradient function $\frac{dy}{dx} = 3x^2 + 4x - 1$. Find the co-ordinates of the point on the curve where $x = -1$.
A small business makes and sells candles. The rate of change of its weekly profit, $P$ dollars, with respect to the number of candles sold, $x$, is
$\frac{dP}{dx} = 40 - 0.4x$
When no candles are sold, the business makes a loss of \$200 for the week.
Find the weekly profit when 60 candles are sold.
The second derivative of a function is given by $f''(x) = 6x - 4$.
Given that the gradient of the function at $x = 2$ is 7, and the function passes through the point $(1, 5)$, find the equation of the original function $f(x)$.
The gradient of a function is given by $f'(x) = 2x - 4$. The graph of the original function, $f(x)$, passes through the point $(1, 2)$.
Find the equation of the tangent to the graph of $f(x)$ at the point where $x=3$.
The gradient function of a curve is given by $f'(x) = px + 5$. The curve passes through the points $(1, 4)$ and $(-2, -5)$. Find the equation of the curve.
The derivative of a function is given by $f'(x) = 6x^2 - 6x - 12$. The graph of the function has a local maximum at the point $(-1, 15)$. Use calculus to find the value of the local minimum of the function.
The gradient of a curve is given by
$\frac{dy}{dx} = 3x^2 + 2kx - 9$
where $k$ is a constant. The curve has a turning point where $x = 3$, and it passes through the point $(0, 4)$.
Find the coordinates of the other turning point of the curve, and state whether it is a maximum or a minimum.
Water flows into and out of a tank. The rate of change of the volume of water in the tank is
$\frac{dV}{dt} = 6t - t^2$
litres per minute, where $t$ is the time in minutes. At $t = 0$ the tank holds 20 litres.
Use calculus to find the greatest volume of water in the tank during the first 8 minutes, and justify that it is a maximum.
The graphs of a function $f(x)$ and its derivative $f'(x)$ are shown. Use information from the graphs to find the equation of the function $f(x)$.
The graphs of a function $f(x)$ and its derivative $f'(x)$ are shown. Use calculus and information from the graphs to find the coordinates of the minimum point of the function $f(x)$.
The second derivative of a function is given by $f''(x) = 12x + 6$.
Given that the function passes through the points $(1, 8)$ and $(2, 20)$, find the equation of the original function $f(x)$.