Tangents & Normals at a given point: notes and practice questions

The height of a section of a roller coaster track, in metres, can be modelled by the function $h(x) = \frac{1}{4}x^4 - 2x^2$, where $x$ is the horizontal distance in metres from a central point.

(a) Find an expression for the gradient function of the track, $h'(x)$.

(b) At a specific point on the track, where $x=1$, a support beam L is tangent to the track. The coordinates of this point are $(1, -\frac{7}{4})$.

Use your answer to part (a) to find the gradient of the support beam L.

(c) Determine the number of other points on the track where the tangent line is parallel to the support beam L. Justify your answer.

A botanical garden features a winding path for visitors. The path can be modelled by the function $f(x) = x^3 - 3x^2 - 9x + 5$. All distances in the garden are in kilometres.

A new straight maintenance path needs to be constructed. This path will start at point P on the visitor path, which has coordinates $(2, -17)$.

(a) Using your graphic display calculator, find the value of $f'(2)$.

(b) Find the equation of the line normal to $f(x)$ at point P.

(c) The maintenance path connects point P to another point Q on the visitor path, where the normal line intersects the path again. For safety regulations, point Q must have a positive x-coordinate. Determine the length of this new maintenance path.

A designer is creating a decorative curve for a new architectural feature. The profile of the curve can be modelled by the function $f(x) = x^3 + \frac{4}{x}$, for $x \neq 0$.

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

(b) The designer wants to place a support beam perpendicular to the curve at the point where $x=1$. Given that the point on the curve is $(1, 5)$, find the equation of the normal to the curve at this point, in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$.

(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.

(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) (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.

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, $P(x)$.

Emily and Liam are researching the adoption of smart home devices in a specific region to create a model predicting future usage. They collect the following data:

Emily proposes the number of devices can be modelled using quadratic regression to find a function of the form $N(x) = ax^2 + bx + c$, where $x$ is the number of years after 2000.

Find the equation of Emily's model.

Emily finds the coefficient of determination for her model is $0.97977$ to five significant figures.

State whether the coefficient of determination supports Emily's proposal. Justify your answer.

Comment on the validity of Emily's model with reference to one of the parameters in the equation.

(i) Find the value of $N'(25)$ and interpret this value in context.

(ii) By considering the changes in device adoption in the table, use the value found in part (d)(i) to comment on the validity of Emily's model.

Liam proposes that the device adoption instead follows a logistic model of the form

$G(x) = \frac{1500}{1+149 \text{e}^{-0.15x}}$

where $x$ is the number of years after 2000 and $G(x)$ is the number of devices in thousands.

State a reason why it may be valid to use Liam's proposal to predict future device adoption.

(i) Find $G'(x)$.

(ii) Hence find the year, according to Liam's model, during which the greatest device adoption growth rate occurred.

12. [Maximum mark: 7]

The path of a small drone flying over a landscape can be modelled by the curve with equation $y = 3x^2 - \frac{8}{x^2}$, where $x$ is the horizontal distance in meters from a reference point and $y$ is the altitude in meters.

(a) Find $\frac{dy}{dx}$.

(b) Write down the gradient of the path when the drone is at a horizontal distance of $x = 2$ meters.

(c) Hence, find the equation of the normal to the drone's path at $x = 2$.

The vertical position, $y$ meters, of a particle moving along a curved path is given by the equation $y = 3x^3 - \frac{5}{x}$, where $x$ is the horizontal distance in meters.

(a) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

(b) Write down the gradient of the curve at $x = 1$.

(c) Hence, find the equation of the normal to the curve at $x = 1$.

(a) The profile of a section of a mountain trail can be modelled by the function $y = x^3 - 5x^2 + 2x$, where $y$ is the altitude in hundreds of metres and $x$ is the horizontal distance in hundreds of metres. Find the gradient of the trail at the point where the horizontal distance is $x = 3$ (and the altitude is $y = -12$).

(b) A company's profit, $P$, in thousands of dollars, from selling $x$ units of a new product is modelled by the function $P(x) = 2x^3 - 9x^2 + 12x + 5$. Find the number of units $x$ at which the profit's rate of change is zero. State the corresponding profit for each of these values of $x$.

(c) The concentration of a certain chemical in a solution, $C$, measured in milligrams per litre, is modelled by the function $C(t) = 150 + 20t - 0.5t^2$, where $t$ is the time in minutes after the chemical is added. Find the rate of change of the concentration with respect to time at the instant when $t = 10$ minutes.

A company's profit, $P(x)$, in thousands of dollars, from selling $x$ thousand units of a new smart device, is modelled by the function $P(x) = \frac{2x+1}{x^2+3}$, for $x \ge 0$.

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

Find the equation of the tangent to the profit curve at the point where $x = 1$.

Determine the number of units sold (to the nearest unit) that maximizes the company's profit, and state the maximum profit (to two decimal places).

Determine the range of values of $x$ (number of thousands of units sold) for which the company's profit is increasing.

(a) A drone's displacement, $s$ metres, from a fixed charging station after $t$ minutes (where $t \geq 0$) is given by the equation $s(t) = \frac{t^2}{t+1} + \ln(t+1)$.

Calculate the distance travelled by the drone in the first 3 minutes.

(b) Find an expression for the velocity, $v(t)$, of the drone. Hence, determine if the drone ever becomes stationary for $t \geq 0$.

(a) A company's profit, $P$, in thousands of dollars, from selling $x$ units of a new product can be modelled by the function $P(x) = x^3 - 3x^2 - 9x + 5$, for $x \ge 0$.

Calculate the coordinates of the stationary points of the profit function $P(x)$.

(b) Find the equation of the normal to the curve $y = P(x)$ at the point where $x = 1$. Give your answer in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$.

(c) Determine the second derivative of the profit function, $P''(x)$.

(d) Use the second derivative to classify the nature of the stationary points found in part (a).