Straight line (gradient-int, general, point gradient form): notes and practice questions

Consider the line $L_{1}$ to be a normal to the graph of $f(x)$ at the point $D( - 1,1).$ Consider the line $L_{2}$ $y = - \frac{2}{3}x + \frac{1}{3}$ to be the tangent to the graph of $f(x)$ at point D.

$a$ Find the gradient of $L_{1}$.

$b$ Find the equation of $L_{1}$, giving the answer in the form $ax + by + d = 0$, where $a$, $b$ and $d\mathbb{\in Z}$.

Consider the function $f(x) = \frac{9}{x} + 2x^{2} - 9$, $x \neq 0$.

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

$b$ Find the coordinates of the point at which the normal has a gradient of $- \frac{1}{11}\ .$

Three emergency service hubs are located at coordinates F1 (2, 5), P1 (10, 5), and H1 (10, 15), where distances are measured in kilometers. A central dispatch unit needs to be built at a location equidistant from these three hubs. This location is a vertex of the Voronoi diagram formed by the hubs.

Write down the coordinates of this central dispatch unit.

Find the equation of the perpendicular bisector of the segment connecting F1 (2, 5) and H1 (10, 15).

A new emergency hub, S1 (4, 8), is established, and the Voronoi diagram is redrawn. Several potential locations for a new drone charging station are at the vertices of this updated diagram. The coordinates of four such vertices are given as V_alpha (6, 10), V_beta (6, 4.5), V_gamma (8.75, 10), and V_delta ($\frac{223}{28}$, $\frac{59}{7}$). The drone charging station should be built at the vertex that is as far as possible from its nearest emergency service hub.

By calculating appropriate distances, find the location of the drone charging station.

Hence, write down the distance of the drone charging station to the nearest emergency service hub.

Consider the function $y = \left( x^{3} - 2x^{2} \right)e^{x}$.

(a) Find $\frac{d^{2}y}{dx^{2}}$.

(b) Find how many points of inflection does this function contain.

A company is designing an open-top storage container with a square base. The side length of the base is $x$ cm and the height is $h$ cm. The container needs to have a volume of $500$ cm$^3$.

Explain why $x^2 h = 500$.

Rearrange the equation in part (a) to make $h$ the subject.

Write down an expression for the surface area, $A$, of the open-top container.

Show that this can be written as

$A = x^2 + \frac{2000}{x}$

Plot the graph of $A = x^2 + \frac{2000}{x}$ for $x > 0$.

Find the minimum surface area and the value of $x$ when this occurs.

Consider the function $y = \left( x^{3} - 2x^{2} \right)e^{x}$.

$a$ Find $\frac{d^{2}y}{dx^{2}}$.

$b$ Find how many points of inflection does this function contain.

(a) An architect is designing a decorative archway for a park entrance. The cross-section of one half of the archway is modeled. The archway is symmetrical about the y-axis.

The architect models the base section of the archway as a straight line passing through the points $(0, 2)$ and $(2, 4)$, where all units are in metres.

Find the equation of the line passing through these two points.

(b) The architect initially models the curved upper section of the archway using the following measured points:

$(2, 4)$, $(4, 5)$, $(5.5, 3)$, and $(7, 0)$.

(i) Find the equation of the least squares regression quadratic curve for these four points.

(ii) By considering the gradient of this curve when $x = 2$, explain why it may not be a good model for the archway.

(c) The architect decides that a better model for the curved section would be a quadratic curve with a maximum point at $(4.5, 5.5)$ and that passes through the endpoint $(7, 0)$.

Find the equation of this new quadratic model.

(d) Believing this to be a better model for the archway, the architect wants to estimate the volume of the solid generated by rotating this half-archway about the x-axis.

(i) Write down an expression for this estimate of the volume as a sum of two integrals.

(ii) Find the value of this estimate.

The diagram shows the slope field for the differential equation $\frac{dy}{dx} = \cos(x-y)$ for $-\pi \le x \le \pi$ and $-\pi \le y \le \pi$.

The local maximum points for solutions to the differential equation lie on the straight line $L_1$.

Find the equation of $L_1$, giving your answer in the form $y = mx + c$.

Find the equation of the straight line $L_2$ on which all local minimum points lie within the given domain, giving your answer in the form $y = mx + c$.

A geological survey team is setting up sensors in a remote area. A map of the area is represented on the following coordinate axes.

Three sensor locations are positioned at $S_1(0, 10)$, $S_2(16, 14)$ and $S_3(16, 0)$.

All measurements are in kilometres.

(a) Write down the distance between $S_2$ and $S_3$.

(b) Calculate the distance between $S_1$ and $S_2$.

(c) A geological team member is at sensor $S_2$ and needs to walk directly to sensor $S_1$. Calculate the bearing of $S_1$ from $S_2$.

A communication relay station is to be installed at a point that is an equal distance from each of the sensors at $S_1$, $S_2$, and $S_3$.

(i) Write down the gradient of the line segment $[S_1S_3]$.

(ii) Write down the coordinates of the midpoint of the line segment $[S_1S_3]$.

(iii) Hence, calculate the coordinates of the communication relay station.

A security system is being set up in a warehouse. Laser beams are emitted from various points. On a coordinate plane, where one unit represents 1 meter, the positions of three sensors are given by A(1, 2), B(0, 6), and C(5, 8). A fourth sensor, D, is located on the x-axis.

The main control unit ensures that the path from A to C is perpendicular to the path from B to D.

Find the gradient of the line segment connecting sensor A and sensor C.

Write down the gradient of the line segment connecting sensor B and sensor D.

Find the equation of the line passing through sensor B and sensor D. Give your answer in the form $ax + by + d = 0$, where $a, b$ and $d$ are integers.

Write down the x-coordinate of sensor D.

(a) (i) Consider the function $f(x) = x^3$. Find $f'(x)$.

(ii) The first section of the stone wall's profile is given by $y = f(x)$ for $0 \le x \le 0.5$. A straight glass panel is to be installed tangent to this section of the wall at the point where $x=0.5$. Find the equation of this tangent line.

The full profile of the stone wall, $F(x)$, is defined by:

$F(x) = \begin{cases} x^3 & 0 \le x \le 0.5 \\ 0.75x - 0.25 & 0.5 < x \le 1.0 \end{cases}$

A smaller, decorative stone insert is designed using a transformation of $F(x)$. The graph of $G(x)$ is obtained from the graph of $F(x)$ by:

• a stretch scale factor of $\frac{1}{2}$ in the $x$ direction,

• followed by a stretch scale factor of $\frac{1}{2}$ in the $y$ direction,

• followed by a translation of $0.5$ units to the right.

Point P lies on the graph of $F(x)$ and has coordinates $(1.0, 0.5)$. Point Q is the image of P under the given transformations and has coordinates $(q_x, q_y)$.

Find the value of $q_x$ and the value of $q_y$.

The piecewise function $G(x)$ is given by

$G(x) = \begin{cases} k(x) & c \le x \le d \\ mx + n & d < x \le q_x \end{cases}$

(c) Find

(i) an expression for $k(x)$.

(ii) the value of $d$.

(iii) the value of $n$.

(d) (i) Calculate the total area of the profile of the stone wall, enclosed by $y = F(x)$, the $x$-axis, and the line $x = 1.0$.

The decorative insert $G(x)$ is placed within the main wall profile $F(x)$. The region of the main wall profile that is not covered by the insert is to be painted a contrasting colour. This region is bounded by $y=F(x)$, the $x$-axis, and the lines $x=0$ and $x=1$, excluding the area under $G(x)$ from $x=0.5$ to $x=1.0$. Find the area of this region.

Two observation posts, Alpha (A) and Bravo (B), are located in a national park. Their positions are given by coordinates in kilometres relative to a central ranger station. Post A is at $(-4, 2)$ and Post B is at $(6, 8)$. A new patrol route is established along the line that is equidistant from both posts. This route is represented by the perpendicular bisector of the line segment [AB].

(a) Find the equation of the line that the patrol route follows.

A supply drop point, Charlie (C), is located on the patrol route. Post C is due east of Post A.

(b) Find the x-coordinate of Post C.

A drone delivery service operates from three main hubs, P, Q, and R, whose locations are represented on a coordinate plane. All measurements are in kilometres.

Hub P is at $(0, 10)$, Hub Q is at $(8, 14)$, and Hub R is at $(8, 2)$.

Write down the distance between Hub Q and Hub R.

Calculate the distance between Hub P and Hub Q.

A drone is at Hub Q and needs to fly directly to Hub P. Calculate the bearing of P from Q.

A central charging station is to be built at a location equidistant from Hubs P, Q, and R.

Write down the gradient of the line segment [PR].

Find the coordinates of the midpoint of [PR].

Hence calculate the coordinates of the central charging station.

Two historical landmarks are located at points A(4, 10) and B(16, 2) on a coordinate map. A proposed new scenic path, represented by the line R with equation $y = x - 2$, passes near these landmarks. A new visitor centre is to be built on this scenic path such that it is equidistant from both landmarks.

(a) Find the equation of the perpendicular bisector of the line segment [AB]. Give your equation in the form $y = mx + c$.

(b) Determine the coordinates of the point on the scenic path R where the visitor centre should be located.

(a) SweetTreats is designing new packaging for a line of artisanal chocolates. The initial design is a box in the shape of a cuboid with a square base of side length $L$ cm. Its height, $H$ cm, is twice the length of the base.

Write down an expression for $H$ in terms of $L$.

(b) The box is designed to hold $250$ cm$^3$ of chocolates.

Find the value of $L$ and $H$.

(c) Calculate the total external surface area of the box.

(d) To minimize the amount of material needed, SweetTreats is considering changing the shape to a cylinder with radius $r$ cm and height $h$ cm. The cylindrical container must also hold $250$ cm$^3$ of chocolates.

Find an expression for the height, $h$, of the container in terms of $r$.

(e) Let the total external surface area of the cylindrical container be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{\text{d}A}{\text{d}r}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ needed for the cylinder.

(h.i) Find $\frac{\text{d}^2 A}{\text{d}r^2}$.

(h.ii) Hence determine whether the graph of $A$ is concave-up or concave-down for $r > 0$. Justify your answer.

(a) Write down the gradient of the signal path $S_1$.

(b) Find the equation of the signal path $S_1$ in the form $y = mx + c$.

(c) Show that $S_2$ is not the line that is normal to $f(x)$ at point $P$.

A botanical garden is researching the growth of a rare orchid species, Orchidaceae splendens, in a controlled greenhouse environment. They are investigating the long-term viability of maintaining a large population.

The population of Orchidaceae splendens, $P$, measured in thousands, is shown in the following table. The time $t$ is measured in years from the beginning of 2010, where $t \in \mathbb{R}$.

The garden models this data set using the logistic function $P = \frac{250}{1+Ce^{-kt}}$, where $C, k \in \mathbb{R}^+$.

In context, explain the significance of $250$ in this logistic function.

Use the value of $P$ at $t = 1$ to show that $C= 499e^k$.

Use the value of $P$ at $t = 7$ to find a second expression for $C$.

Use your answers to part (b) to find a value for $k$.

Use your answers to part (b) to find a value for $C$.

The botanical garden uses its model to predict values of $P$ at the end of the years 2011 to 2015. These values are shown in the following table, correct to one decimal place.

Calculate the value of $a$ correct to one decimal place.

Using the value of $a$ correct to one decimal place, find the sum of square residuals, $SS_{res}$, when using this model to predict the values of $P$.

As a measure of how well the model fits the data, the botanical garden uses the error function, $E$, where $E = \sqrt{\frac{SS_{res}}{n}}$, and $n$ is the number of predictions made using the model.

The garden decides they will use this model if $E$ is less than $0.15$.

By finding the value of $E$ for the model, show that the garden will decide that the model can be used.

Use the model to predict the population of Orchidaceae splendens in the greenhouse at the end of 2025.

One of the challenges in maintaining the orchid population is providing sufficient specialized protective covers. The botanical garden estimates that $15\%$ of orchids will require a specialized protective cover, and each cover can protect $500$ individual orchids.

Use the model to find an expression for the total number of specialized protective covers required at time $t$. Give your answer in thousands of covers.

At the end of 2014 ($t=5$) there were $2.5$ thousand specialized protective covers in the greenhouse, and at the end of 2016 ($t=7$) there were $3.5$ thousand.

Find the average number of specialized protective covers installed per year during this period.

The botanical garden assumes that the number of specialized protective covers will continue to increase linearly, at the same rate, until the population stabilizes.

Determine the value of $t$, where $t \ge 5$, at which the number of specialized protective covers will first be insufficient to meet demand, and hence the year in which this occurs.

Three sensor stations, Alpha (A), Beta (B), and Gamma (C), are positioned in a national park. Their coordinates are A(2, 8), B(10, 4), and C(6, 0) respectively.

The diagram below shows these points and the perpendicular bisectors of the segments connecting them.

The perpendicular bisector of the line segment [BC] intercepts the axes at coordinates (0, 10) and (10, 0).

Write down the equation of the perpendicular bisector of [BC].

The equation of the perpendicular bisector of [AB] is $y = 2x - 6$.

Find the coordinates of point V, where the three perpendicular bisectors meet. Give your answer to four significant figures.

A Voronoi diagram is constructed with sensor stations A, B, and C as the three sites.

Draw, clearly, the edges of the Voronoi diagram on the given diagram.

A robotics engineer is programming a swarm of drones. The initial paths of the drones are given by a family of lines, $L_k$, with equation $y = kx + (5-2k)$, where $k$ is a real parameter and $k \neq -1.5$.

A control signal applies a linear transformation to the coordinates of every drone, described by the matrix $T = \begin{pmatrix} 3 & 2 \\ -6 & -4 \end{pmatrix}$.

The new path of a drone is the line $L'_k$.

(a) Find a vector equation for the line $L_k$ in terms of $k$.

(b) Find the determinant of $T$.

(c) Show that all the drones will end up moving along the same path, by showing that the equation of the transformed line $L'_k$ does not depend on $k$.

A landscape architect is designing a walking path in a new park. The path consists of three segments. The first segment is a straight path connecting point A(0, 0) to point B(2, 4).

Write down the equation of the line segment for $0 \leq x \leq 2$.

A curved section of the path, modeled by a quadratic function, connects point C(-2, 2) to point A(0, 0). At point A(0, 0), the curve has the same gradient as the straight path segment AB.

Find the equation of the curve between (-2, 2) and (0, 0).

The second curved section of the path, also modeled by a quadratic function, connects point B(2, 4) to point D(5, 1). At point B(2, 4), the curve has the same gradient as the straight path segment AB.

Find the equation of this curve.

Write down the equation of the entire walking path as a piecewise function, $P(x)$.