L2 Calculus
Question Bank
Use calculus to determine if the function $f(x) = 2x^3 - 9x^2 + 12x - 5$ is increasing or decreasing at the point $(2, 3)$.
Use calculus to determine whether the curve $y = (x - 2)(x + 3)$ is increasing or decreasing at $x = 1$.
Determine whether the function $g(x) = \frac{x^3 - 6x^2}{x}$ is increasing or decreasing when $x = -1$.
The height of a ball thrown in the air is given by $h(t) = -5t^2 + 20t + 2$, where $h$ is the height in metres and $t$ is the time in seconds.
(a) Is the height increasing or decreasing at $t = 1$ second?
(b) Is the height increasing or decreasing at $t = 3$ seconds?
For what values of $x$ is the function $f(x) = x^3 + 2x^2 - 5x + 1$ a decreasing function?
For what values of $x$ is the function $f(x) = -\frac{1}{3}x^3 - x^2 + 8x + 1$ an increasing function?
The function $f(x) = x^3 - kx$, where $k$ is a positive constant, is a decreasing function only for the interval $-2 < x < 2$. Use calculus to find the value of $k$.
The price of one share in a company, $P$ dollars, $t$ months after the start of the year is modelled by
$P = t^3 - 15t^2 + 63t + 20$
for $0 \le t \le 12$.
An investor claims:
"The share price fell for exactly one-third of the year, and while it was falling it lost almost a third of its value."
Use calculus to evaluate both parts of the investor's claim.
Consider the graph of $f(x)$ $= -k^2x^3 + 6kx^2 - 9x + 10$, where $k$ is a positive constant. Find expressions, in terms of $k$, for the range(s) of values of $x$ for which this graph is decreasing.
Find the values of $k$ for which the function
$f(x) = x^3 + kx^2 + 12x$
is never decreasing.