L2 Calculus
Question Bank
Use calculus to verify that $x = 2$ is a stationary point of the curve $y = x^3 - 6x^2 + 12x - 5$.
Use calculus to determine whether $x = 4$ is a stationary point of the curve $y = x^2 - 6x + 5$.
Find the coordinates and nature of the stationary points of the curve $y = x^3 - 3x^2 - 9x + 5$.
The graph of a function $f(x) = x^3 + bx^2 + 5$ has a turning point when $x=4$. Find the value of $b$.
A company's weekly profit (in thousands of dollars) is modelled by $P(x) = -2x^2 + 16x - 10$, where $x$ is the number of units produced (in hundreds).
The company manager claims that producing 400 units per week (i.e., $x = 4$) will give a stationary point for profit.
Use calculus to verify whether the manager is correct.
The curve $y = x^3 - 6x^2 + kx + 5$ has a stationary point at $x = 3$, where $k$ is a constant.
Use calculus to find the value of $k$.
A section of a mountain-bike track is modelled by
$h = -0.002x^3 + 0.06x^2 + 1$
for $0 \le x \le 30$, where $h$ is the height of the track in metres and $x$ is the horizontal distance in metres from the start of the section.
Use calculus to find the coordinates of the highest point on this section of track, and justify that it is a maximum.
Show that the curve
$y = x^3 - 6x^2 + 15x - 4$
has no stationary points.
Find the coordinates and nature of the stationary points of the curve $\small y = x^4 - 8x^3 + 22x^2 - 24x + 12.$
Use calculus to find the quadratic function $f(x)$ that has the following properties:
- a turning point when $x = 3$
- a gradient of -8 when $x = 1$
- a value of -5 when $x = 5$
The function $f(x) = x^4 + ax^3 + bx^2 + 3$ has turning points when $x = 1$ and $x = -2$. Find the values of $a$ and $b$.
Use calculus to prove that the graph of the quartic function $y = x^2(x-4)^2$ has a local maximum when $x=2$.
The graph of the function $y = x^3 - 3x^2 + kx + 10$ has a turning point at $x = 4$. Find the coordinates of both turning points and determine their nature.
A cubic function, $f(x) = ax^3 + bx^2 + cx + d$, has turning points that occur at $x = -2$ and $x = 4$. Find an expression for $b$ in terms of $a$.