Limits & derivative definition, increasing / decreasing functions: notes and practice questions

A function is defined by $f(x) = x^2 - 6x + c$, where $x, c \in \mathbb{R}$. The graph of $f$ is tangent to the line $L$ with equation $y = 2x - 11$.

(a) Show that $c=5$.

(b) The function $f$ can be expressed in the form $f(x) = (x-p)(x-q)$, where $p, q \in \mathbb{R}$.

Find the value of $p$ and the value of $q$.

(c) The function $f$ can also be expressed in the form $f(x) = (x-h)^2 + k$, where $h, k \in \mathbb{R}$.

Find the value of $h$ and the value of $k$.

(d) Hence find the values of $x$ where the graph of $f$ is both positive and decreasing.

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 $B$.

A small reconnaissance drone is performing a vertical ascent and descent. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, for $0 \le t \le 10$, is modelled by the function $v(t) = t \sin t - 2.5$.

The following diagram shows the graph of $v$.

(a) Find the smallest value of $t$ for which the drone is momentarily stationary.

(b) Find the total vertical distance travelled by the drone during the first 10 seconds.

(c) Find the vertical acceleration of the drone when $t=8$ seconds.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

A mini-submarine is performing a test dive. Its velocity, $v$ in m/s, at time $t$ seconds is given by $v(t) = 5\sin(t) - t + 2$, for $0 \le t \le 8$.

(a) Calculate the time(s) when the submarine is momentarily at rest.

(b) Find the acceleration of the submarine when it first changes direction.

(c) Determine the total distance travelled by the submarine during the first 8 seconds of its dive.

(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 amount of a certain pollutant, in tonnes, in a lake $t$ weeks after a major clean-up operation began, can be modelled by $P(t) = 5 + 75e^{-0.1t}$, $t \ge 0$.

Find the initial amount of pollutant in the lake.

Find the amount of pollutant in the lake five weeks after the clean-up operation began.

Write down the value of $P'(5)$.

Interpret the meaning of your answer to part (b) in the given context.

The clean-up operation is considered successful when the amount of pollutant in the lake is below 20 tonnes.

Find the least possible integer value of $k$, in weeks, for which the amount of pollutant is below 20 tonnes.

As the clean-up operation continues indefinitely, the amount of pollutant in the lake approaches a constant level.

Find this constant level of pollutant.

Find the limit of $P'(t)$ as $t$ approaches infinity.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

A buoy floats on the surface of a lake. Its vertical displacement, $s$ metres, from its equilibrium position at time $t$ seconds is modelled by the function $s(t) = 5 \cos(2t+1)$, for $0 \leq t \leq 5$.

(a) Find the first time, $q$ seconds, when the buoy momentarily comes to rest.

(b) Find the total distance that the buoy travels in the first $q$ seconds.

Consider the limit $\lim_{x\to0} \frac{\ln(\cos x + 1) -k}{x^2}$, where $k \in \mathbb{R}$.

Show that a finite limit only exists for $k = \ln 2$.

Using l'Hôpital's rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

A drone's altitude, $h$ metres, above the ground at time $t$ seconds is modelled by the function $h(t) = 8 \cos(0.5t + 1) + 12$, for $0 \leq t \leq 15$.

(a) Find the first time, $q$ seconds, when the drone's vertical velocity is zero.

(b) Find the total vertical distance that the drone travels in the first $q$ seconds.

The concentration of a specific chemical, $C$, in a reaction vessel (in moles per litre) after $t$ minutes is modelled by the function $C(t) = \frac{25(1+0.5t)}{4+0.02t}$.

Write down the initial concentration of the chemical in the reaction vessel.

Calculate the concentration of the chemical in the reaction vessel after 10 minutes.

Calculate the number of minutes that will have passed when the chemical concentration reaches 200 moles per litre.

Show that, when governed by this model, the concentration of the chemical in the reaction vessel cannot exceed 625 moles per litre.

(a) Use the principle of mathematical induction to prove that

$\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} = \frac{n}{2n+1}$

for all $n \in \mathbb{Z}^+$.

(b) Hence, find the value of $\sum_{r=5}^{15} \frac{1}{(2r-1)(2r+1)}$.

(c) Hence, show that the infinite series $\sum_{r=1}^{\infty} \frac{1}{(2r-1)(2r+1)}$ converges, and find its sum.

A remote-controlled drone is launched from a platform and moves horizontally in a straight line. Its velocity, $v$ m s$^{-1}$, at time $t$ seconds after launch, is given by $v(t) = t + 1 - \sqrt{4t + 16}$, for $t \ge 0$.

(a) Find the drone's initial velocity.

(b) Find the time when the drone is at rest.

(c) Find the drone's acceleration at the instant it comes to rest.

(d) Determine the time interval(s) when the drone is:

(i) slowing down

(ii) speeding up.

A new drug is administered to a patient, and its concentration, $C$, in the bloodstream (in mg/L) after $t$ hours is modelled by the function $C(t) = \frac{15(1 + 0.48t)}{6 + 0.4t}$.

(a) Write down the initial concentration of the drug in the bloodstream.

(b) Calculate the concentration of the drug in the bloodstream after 5 hours.

(c) Determine the number of hours that will have passed when the drug concentration reaches 12 mg/L.

(d) Show that, according to this model, the drug concentration in the bloodstream can never exceed 18 mg/L.

Consider the function $f(x) = (kx-1)e^{-x}$, for $x \in \mathbb{R}$, where $k$ is a positive real constant.

(a) Find the zero of $f$.

(b) Find the intervals where $f$ is increasing and where it is decreasing.

(c) Find the range of $f$.

(d) Find the intervals where the graph of $f$ is concave up and where it is concave down.

Let $g(x) = kx^2 - \ln(x)$, for $x>0$, where $k$ is a positive constant.

(a) Find expressions for $g'(x)$ and $g''(x)$.

(b) Find, in terms of $k$, the interval(s) where $g$ is increasing and the interval(s) where $g$ is decreasing.

(c) Show that the graph of $g$ is concave up for all values of $x$ in its domain.

Find the following limits, if they exist.

(a) $\lim_{x \to -1} (x^3 - 2x^2 + x - 5)$

(b) $\lim_{x \to 4} \frac{2x - 1}{x^2 - 3}$

(c) $\lim_{x \to 0} \sqrt{9 - 4x^2}$

(d) $\lim_{x \to -2} \frac{x^2 + 5x + 6}{x+2}$

(e) $\lim_{x \to -\infty} \frac{3x}{\sqrt{4x^2 + 1}}$

(f) $\lim_{x \to \infty} \frac{5x^2 - 3}{x^3 + 2x}$

Consider the following three graphs of functions, labelled Graph 1, Graph 2, and Graph 3.

For each condition below, state which graph (1, 2, or 3) satisfies the condition, providing a reason for your choice.

(a) The function $f$ has the property that $f'(x) = 0$ for exactly two distinct values of $x$.

(b) The function $g$ has the property that $g'(x) > 0$ for all $x$ in its domain.

(c) The function $h$ has the property that $\lim_{x \to \infty} h(x) = k$, where $k$ is a finite positive constant.

(d) The function $k$ has the property that $k'(x) < 0$ and $k''(x) > 0$ for all $x$ in its domain.