Special functions derivatives + chain/product/quotient rules: notes and practice questions

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n (x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{(xe^{-x} - x)^2}{x^4}$.

Consider the function $f(x) = e^{2x} - \ln(x)$ and the function $g(x)$ which is obtained through horizontally translating $f(x)$ 1 unit to the right also vertically translating it by 2 units downwards.

$a$ Find $g(x)$.

$b$ The function $g(x)$ has a minimum at $x = a$. Find $a$.

The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = -2x + 9$ is the tangent to the graph of $f$ at $x = 3$.

(a) Write down the value of $f'(3)$.

(b) Find $f(3)$.

The function $g$ is defined for all $x \in \mathbb{R}$ where $g(x) = x^2 - 1$ and $h(x) = f(g(x) )$.

(c) Find $h(2)$.

(d) Hence, find the equation of the tangent to the graph of $h$ at $x = 2$.

Let $f(x) = \frac{1}{\sqrt{1-2x}}$ for $x < \frac{1}{2}$.

(a) Show that $f''(x) = 3(1-2x)^{-\frac{5}{2}}$.

(b) Use mathematical induction to prove that $f^{(n)}(x) = \frac{(2n)!}{2^n n!} (1-2x)^{-\frac{2n+1}{2}}$ for $n \in \mathbb{Z}, n \ge 2$.

Let $g(x) = \ln(1+kx)$, where $k$ is a real constant.

Consider the function $h$ defined by $h(x) = f(x) \times g(x)$ for $x < \frac{1}{2}$.

It is given that the coefficient of the $x^2$ term in the Maclaurin series for $h(x)$ is $-4$.

(c) Find the possible values of $k$.

Consider the functions $f(x) = \ln(x-p)$ and $g(x) = \frac{1}{4}x^2 + q$ where $p, q \in \mathbb{R}$.

(a) Find $g'(x)$.

The graphs of $f$ and $g$ have a common tangent at the point where $x = 2$.

(b) Show that $p = 1$.

(c) Hence, find the value of $q$.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

Let $f(x) = \frac{1}{\sqrt{1+2x}}$ for $x > -\frac{1}{2}$.

(a) Show that $f''(x) = 3(1+2x)^{-\frac{5}{2}}$.

(b) Use mathematical induction to prove that $f^{(n)}(x) = (-1)^n \frac{(2n)!}{2^n n!} (1+2x)^{-\frac{2n+1}{2}}$ for $n \in \mathbb{Z}, n \ge 1$.

Let $g(x) = e^{mx}, m \in \mathbb{R}$.

Consider the function $h$ defined by $h(x) = f(x) \times g(x)$ for $x > -\frac{1}{2}$.

It is given that the $x^2$ term in the Maclaurin series for $h(x)$ has a coefficient of $3$.

(c) Find the possible values of $m$.

The function $f$ is defined by $f(x) = e^x \sinh x$, where $x \in \mathbb{R}$.

Find the Maclaurin series for $f(x)$ up to and including the $x^3$ term.

Hence, find an approximate value for $\int_0^1 e^{x^2} \sinh(x^2)dx$.

The function $g$ is defined by $g(x) = e^x \cosh x$, where $x \in \mathbb{R}$.

Show that $g''(x) = 2g'(x)$.

Hence, find the values of $g'''(0)$ and $g^{(4)}(0)$.

Using the result from part (c), find the Maclaurin series for $g(x)$ up to and including the $x^4$ term.

Hence, or otherwise, determine the value of $\lim_{x \to 0} \frac{2e^x \cosh x - 2 - 2x - 2x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n(x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, find the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^5$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{xe^{-x} - x + x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}\left(\frac{x}{1-x}\right) = n!(1-x)^{-(n+1)}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = \frac{x}{1-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{\left(\frac{x}{1-x} - x\right)^2}{x^4}$.

The functions $f$ and $g$ are defined by $f(x) = \sin x$ and $g(x) = \cot x$, for $0 < x < \frac{\pi}{2}$.

The curves $y = f(x)$ and $y = g(x)$ intersect at a point P whose x-coordinate is $k$.

Show that $\sin^2 k = \cos k$.

Hence, show that the tangent to the curve $y = f(x)$ at P and the tangent to the curve $y = g(x)$ at P are perpendicular.

Find the value of $\cos k$. Give your answer in the form $\frac{a+\sqrt{b}}{c}$, where $a, c \in \mathbb{Z}$ and $b \in \mathbb{Z}^+$.

A ladder must be placed against a tall vertical building, clearing a monument that is 8 m high and stands on horizontal ground 1 m away from the building's base. The ladder touches the ground, the top corner of the monument, and the wall of the building.

Let $L$ be the length of the ladder in metres.

Let $\theta$ be the angle that the ladder makes with the ground, where $0 < \theta < \frac{\pi}{2}$.

(a) Show that $L = \sec \theta + 8\csc \theta$.

(b) (i) Find $\frac{dL}{d\theta}$.

(b) (ii) When $\frac{dL}{d\theta} = 0$, show that $\tan\theta = 2$.

(c) (i) Find $\frac{d^2L}{d\theta^2}$.

(c) (ii) When $\tan\theta = 2$, find the value of $\frac{d^2L}{d\theta^2}$.

(d) (i) Hence, justify that $L$ is a minimum when $\tan\theta = 2$.

(d) (ii) Determine this minimum value of $L$.

(e) A construction company only has ladders with a maximum length of 11 m. Determine whether it is possible to position a ladder against the building over the monument, giving a reason for your answer.

Consider the functions $f(x) = k - (x-h)^2$ and $g(x) = \ln(x-1) + 2$ where $h, k \in \mathbb{R}$.

The graphs of $f$ and $g$ have a common tangent at $x=2$.

(a) Find $g'(x)$.

(b) Show that $h = \frac{5}{2}$.

(c) Hence, find the value of $k$.

A rectangle is inscribed in an ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The sides of the rectangle are parallel to the coordinate axes. The vertices of the rectangle are located at $(\pm x, \pm y)$, where $x > 0$ and $y > 0$.

(a) Show that the area of the rectangle, $A$, can be expressed as $A = \frac{12x}{5}\sqrt{25-x^2}$.

(b) Show that $\frac{dA}{dx} = \frac{12(25-2x^2)}{5\sqrt{25-x^2}}$.

(c) Hence, find the exact dimensions of the rectangle with the maximum possible area.

Consider the function f defined by $f(x) = \ln(x^2 - 3)$ for $x > \sqrt{3}$.

The following diagram shows part of the graph of f which crosses the x-axis at point A, with coordinates $(p, 0)$. The line L is the tangent to the graph of f at the point B.

(a) Find the exact value of $p$.

(b) Given that the gradient of L is $1$, find the x-coordinate of B.

Consider the functions $f(x) = \sin x$ and $g(x) = \cos 2x$, where $0 \le x \le \pi$.

The graphs of $f$ and $g$ are shown in the following diagram.

The graphs intersect at points P and Q. The region enclosed by the two graphs is shaded and labelled R.

(a) Find the $x$-coordinates of P and Q.

(b) Find the area of R.