Solving inequalities (graphically & analytically): notes and practice questions

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 scientist is modeling the behavior of a particle in a fluctuating electromagnetic field. The particle's position over time can be described by a quadratic equation involving a field strength parameter $p$. The equation is given by $px^2 - (p + 7)x + 4p + 28 = 0$, where $p \in \mathbb{R}$.

(a) Write down an expression for the product of the roots of this equation, in terms of $p$.

(b) Hence or otherwise, determine the values of $p$ such that the equation has one positive and one negative real root.

A function $f$ is defined by $f(x) = \frac{4x-1}{2x+3}$, where $x \in \mathbb{R}, x \neq -\frac{3}{2}$.

The graph of $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider the function $g(x) = mx - \frac{1}{3}$, where $m \in \mathbb{R}, m \neq 0$.

(i) Write down the number of solutions to $f(x) = g(x)$ for $m < 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$ for which $f(x) = g(x)$ has two distinct solutions for $x \le 0$.

The graphs of the functions $f(x) = kx^2 - 3x$ and $g(x) = x - k$ intersect at two distinct points.

(a) Find the set of possible values for $k$.

(b) Consider the case when $k=1$. The x-coordinates of the intersection points can be written in the form $x = m \pm \sqrt{n}$, where $m, n \in \mathbb{Z}$. Find the values of $m$ and $n$.

A civil engineer is analyzing the structural integrity of a new bridge design. The deflection of a certain point on the bridge, $D(x)$, in millimeters, is modeled by the function $D(x) = \frac{2x+1}{x^2-4}$, where $x$ represents the horizontal distance in meters from a central support. The model is valid for $x \in \mathbb{R}$, $x\neq p$, $x\neq q$.

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

Find an expression for $D'(x)$.

The graph of $y = D(x)$ has exactly one point of inflexion.

Find the x-coordinate of the point of inflexion.

Sketch the graph of $y = D(x)$ for $-4 \leq x \leq 4$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima (if they exist), and giving the equations of any asymptotes.

Consider a related model for stress distribution, $S(x) = \frac{x^2-4}{2x+1}$ for $x \in \mathbb{R}$, $x \neq -\frac{1}{2}$.

Find the equations of all the asymptotes on the graph of $y = S(x)$.

The engineer needs to identify the regions where the bridge deflection $D(x)$ is less than $1$ mm. Solve $D(x) < 1$ for $x \in \mathbb{R}$.

Consider the functions $f(x) = -3\cos x + 5$ and $g(x) = -3\cos\left(x+\frac{\pi}{2}\right) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, p)$.

Given that the maximum value of $g(x)$ is less than or equal to 1, find the largest possible value of $p$.

The sum of the first $n$ terms of a geometric sequence is given by $S_n = \sum_{k=1}^{n} 3000 \left( \frac{3}{4} \right)^k$. This sequence models the amount of plastic (in kg) removed from a river by an environmental initiative each month.

(a) Find the amount of plastic removed in the first month, $u_1$.

(b) Calculate the total amount of plastic the initiative expects to remove if it continues indefinitely.

(c) The initiative aims to remove an amount of plastic that is within $0.05 \text{ kg}$ of its total expected removal goal. Find the least number of months, $n$, for which the total amount of plastic removed, $S_n$, is within $0.05 \text{ kg}$ of the total expected removal goal.

The functions $f$ and $g$ are defined by

$f(x) = 2\ln x$, where $x > 0$

$g(x) = \ln(k(x-2) )$, where $x > 2, k \in \mathbb{R}^+$.

The graphs of $y = f(x)$ and $y = g(x)$ intersect at two distinct points.

(a) State the equation of the vertical asymptote to the graph of $y = g(x)$.

(b) (i) Show that, at the points of intersection, $x^2 - kx + 2k = 0$.

(b) (ii) Hence show that $k^2 - 8k > 0$.

(b) (iii) Find the range of possible values of $k$.

The graphs intersect at $x=p$ and $x=q$, where $p<q$.

(c) In the case where $k=10$, find the value of $q-p$. Express your answer in the form $a\sqrt{b}$, where $a, b \in \mathbb{Z}^+$.

A team of engineers is testing two autonomous robots, Alpha and Beta, on a straight track. Their positions are measured as the distance from a fixed starting point. The experiment runs for 10 minutes.

The position of Robot Alpha, $P_A$ metres, at time $t$ minutes can be modelled by the function $P_A(t) = 3\sin(2t + 5) + 14t + 20$, where $0 \le t \le 10$.

The position of Robot Beta, $P_B$ metres, at time $t$ minutes can be modelled by the function $P_B(t) = 12t + 25$, where $0 \le t \le 10$.

Use the engineers' models to find the initial position of

(i) Robot Beta;

(ii) Robot Alpha correct to three significant figures.

Find the values of $t$ when Robot Alpha and Robot Beta are at the same position. Give your answers correct to three significant figures.

For $t > 5$, prove that Robot Alpha was always ahead of Robot Beta.

For $0 \le t \le 10$, find the total amount of time when the speed of Robot Beta was greater than the speed of Robot Alpha. Give your answer correct to three significant figures.

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.

A new automated manufacturing process produces components. The time, in minutes, taken for a critical assembly step is modelled by a continuous random variable $X$, with a probability density function defined by

$f(x) = \begin{cases} \frac{2}{\pi \sqrt{9-x^2}} & 0 \leq x \leq 3 \\ 0, & \text{otherwise.} \end{cases}$

Find the exact value of $E(X)$.

Find $P(X < 1.5)$.

The assembly step is considered "efficient" if it takes less than 1.5 minutes. Each assembly step is independent. Determine the least number of assembly steps required to be 99% sure of at least one efficient step.

Ten assembly steps were conducted.

Find the probability that exactly three steps were efficient.

Write down the number of ways these three efficient steps could have occurred consecutively in a batch of 10.

Now consider a batch of $n$ assembly steps where it is given that exactly three efficient steps have occurred.

Write down an expression for the number of ways these three efficient steps could have occurred consecutively.

Find the greatest value of $n$ such that the probability of three consecutive efficient steps is more than 0.05, given that exactly three efficient steps have occurred in the batch.

Consider the arithmetic sequence $c, d, e, ...$, where $c, d, e \neq 0$.

Show that $c + e = 2d$.

Consider the geometric sequence $c, f, g, ...$, where $c, f, g \neq 0$.

Show that $f^2 = cg$.

The first term of both sequences is $c$. It is given that $e = g = 4$ and that $f$ is a real number.

Show that $d > 2$.

Consider the case where $c = 16$, $f > 0$ and $e = g = 4$.

(i) Write down the first four terms of the arithmetic sequence.

(ii) Write down the first four terms of the geometric sequence.

A new sequence $v_n$ is formed by combining the terms of the arithmetic sequence, $A_n$, and the geometric sequence, $G_n$, from part (d). The terms of $v_n$ are given by $v_n = A_n - \log_2(G_n)$.

(i) Show that $v_n$ is an arithmetic sequence and find its common difference.

(ii) Hence, find the value of $\sum_{n=1}^{12} v_n$.

Consider the functions $f(x) = 5 - x^2$, $g(x) = \frac{3}{x-2}$, and $h(x) = e^x + 1$.

(a) Find the range of $f(x)$.

(b) Find the range of $g(x)$.

(c) Find the range of $h(x)$.

(d) Find an expression for $(g \circ f)(x)$.

(e) Solve the equation $(g \circ f)(x) = -1$.

(f) Solve the inequality $(h \circ f)(x) < e+1$.

The depth of water, $D(t)$ metres, in a harbour on a particular day is modelled by the function $D(t) = a \cos(bt) + c$, for $0 \le t \le 12$, where $t$ is the number of hours after midnight.

The graph of $y=D(t)$ is shown below.

The graph has a maximum point at $(0, 12)$ and a minimum point at $(6, 4)$.

(a) State:

i) the range of $D$.

ii) the period of $D$.

(b) Hence, find the values of $a$, $b$ and $c$.

(c) A ship requires a water depth of at least 10 metres to be able to enter the harbour. Find the time intervals, during the first 12 hours, when the ship can enter.

Solve the inequality $3x^2 + 5x - 2 > 0$.

The function $g$ is defined by $g(x) = \sqrt{3x^2 + 5x - 2}$, where $x \in \mathbb{R}, x \ge k$.

Find the least value of $k$ for which $g^{-1}$ exists, justifying your answer.

(a) A deep-sea submersible's vertical displacement, in metres, from a reference depth is given by $s(t) = 5 \sin(2t) - 3 \cos(t)$ for time $t$ minutes. Positive $s(t)$ indicates the submersible is above the reference depth, and negative $s(t)$ indicates it is below.

Find the submersible's vertical velocity and acceleration at any time $t$.

(b) Find the time intervals during $0 \le t \le 2\pi$ when the submersible is moving upwards.

(c) Determine the time intervals during $0 \le t \le 2\pi$ when the submersible's vertical velocity is decreasing.

(d) Calculate the total vertical distance travelled by the submersible during the time $0 \le t \le 2\pi$. Give your answer to three significant figures.

A student claims that $2^n > n^2$ for all integers $n \ge 5$.

(a) Show that $2n^2 > (n+1)^2$ for all integers $n \ge 3$.

(b) Use mathematical induction and the result from part (a) to prove that the student's claim is valid for all integers $n \ge 5$.

A company is designing a new power generator. The power output, $P$, in megawatts (MW), of a prototype generator at time $t$ hours after startup is modelled by the function $P(t) = t^3 - 3ct + K$, where $t \in \mathbb{R}$, $c$ is a control parameter, and $K$ is a constant representing the initial power. For parts (a) to (e), assume $K = 4$.

On separate axes, sketch the graph of $P = P(t)$ showing the value of the $P$-intercept and the coordinates of any points with zero gradient, for

(i) $c = 1$;

(ii) $c = 2$.

Write down an expression for $P'(t)$.

Hence, or otherwise, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) a point of inflexion with zero gradient;

(ii) one local maximum point and one local minimum point;

(iii) no points where the gradient is equal to zero.

Given that the graph of $P = P(t)$ has one local maximum point and one local minimum point, show that

(i) the $P$-coordinate of the local maximum point is $2c^{\frac{3}{2}} + 4$;

(ii) the $P$-coordinate of the local minimum point is $-2c^{\frac{3}{2}} + 4$.

Hence, for $c > 0$, find the set of values of $c$ such that the graph of $P = P(t)$ has

(i) exactly one $t$-axis intercept;

(ii) exactly two $t$-axis intercepts;

(iii) exactly three $t$-axis intercepts.

Consider a modified power output function $Q(t) = t^3 - 3ct + d$ for $t \in \mathbb{R}$ and where $c, d \in \mathbb{R}$. Find all conditions on $c$ and $d$ such that the graph of $P = Q(t)$ has exactly one $t$-axis intercept, explaining your reasoning.