Power rule: notes and practice questions

A company is designing a closed cylindrical container to hold a specific volume of liquid. The total surface area of the container, in cm$^2$, with a fixed volume of $16\pi$ cm$^3$ and a radius of $r$ cm, is given by the function $A(r) = 2\pi r^2 + \frac{32\pi}{r}$, where $r > 0$.

Find $A'(r)$.

Solve $A'(r) = 0$.

Interpret your answer to (b)(i) in context.

The concentration of a reactant A, in mg/L, in a chemical reaction over time $t$ (in minutes) is modelled by the function:

$C(t) = -t^3 + 15t^2 - 48t + 100$, for $0 \le t \le 10$.

(a) Find the coordinates of the local minimum point of the concentration.

(b) Find the coordinates of the local maximum point of the concentration.

(c) Find the set of values of $t$ for which the concentration of reactant A is above $100$ mg/L.

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) A pharmaceutical company is designing a new cylindrical container for a special liquid. The container has a fixed volume $V = 500 \text{ cm}^3$.

The radius of the container is $r \text{ cm}$ and the height is $h \text{ cm}$.

Show that $\pi r^2 h = 500$.

(b) Find an expression for the total surface area $S$ of the container.

(c) Substitute an expression for $h$ (from part (a) ) into your expression for $S$ (from part (b) ) and hence show that $S = 2\pi r^2 + \frac{1000}{r}$.

(d) Find $\frac{dS}{dr}$.

(e) Find the minimum value of $S$ and the values of $r$ and $h$ when this occurs. Show that this value of $S$ is indeed a minimum.

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 drone is launched vertically upwards from a platform. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, is given by the function:

$v = -3t^2 + 18t - 15$, for $t \ge 0$.

Find the times when the drone is momentarily at rest.

Find the magnitude of the drone's vertical acceleration at $t = 5$ seconds.

Find the greatest speed of the drone in the interval $0 \le t \le 5$.

The drone starts from an initial height of $10$ metres above the ground. Find an expression for the height of the drone, $h$ metres, above the ground at time $t$ seconds.

Find the total distance travelled by the drone in the interval $0 \le t \le 4$.

The power output $P$, in watts (W), of a solar panel is modelled by the equation

$P = 250 - 1.5\theta^{0.25}$

where $\theta$ is the angle of inclination of the panel to the sun in degrees. Find an expression for $\frac{dP}{d\theta}$.

At a particular moment, the solar panel is inclined at an angle of 30 degrees, and this angle is increasing at a rate of 2 degrees per minute.

Find the rate of change of the power output at this time.

A tech company, 'CaseCrafters', designs and sells custom phone cases. Their market research suggests that the average number of cases they will sell each month is modelled by the equation

$n = 15000 - 500x$

where $n$ represents the number of cases sold and $x$ represents the selling price, in USD, of each case.

The marketing team proposes selling the cases for $22$ USD each.

Find the average number of phone cases that this model predicts CaseCrafters will sell at this price.

Calculate CaseCrafters' average monthly income, before any expenses, at this selling price.

Hence, write down the function $R(x)$ that can be used to predict CaseCrafters' average monthly income, before expenses, at any selling price, $x$.

CaseCrafters has $8000$ USD of fixed monthly operational costs. Additionally, CaseCrafters must pay their supplier $8$ USD for each phone case.

Calculate CaseCrafters' average monthly profit if they sell each case at a price of $22$ USD.

Show that the average monthly profit for any selling price, $x$, can be found using the function $P(x) = -500x^2 + 19000x - 128000$.

Find $P'(x)$.

Show that the marketing team's selling price of $22$ USD does not maximize their average monthly profit.

CaseCrafters negotiates a new deal with their supplier. Under the new deal, the supplier agrees to discount the cost of each case based on the number of cases purchased by CaseCrafters. The cost charged by the supplier for each case can be found using the function

$C_{\text{supplier}}(n) = 8 - 0.00005n$

where $n$ represents the number of cases sold by CaseCrafters.

Find the function that can be used to find CaseCrafters' average monthly profit using the new deal from the supplier.

Hence, find the selling price, per case, that CaseCrafters should choose in order to maximize their average monthly profit under the new deal.

A company is designing a new cylindrical storage tank. The cost of manufacturing, in thousands of dollars, is modelled by the function $C(r) = r^2 + \frac{100}{r}$, where $r$ is the radius of the tank in metres, and $r > 0$.

(a) Write down the equation of the vertical asymptote of $C(r)$.

(b) Find $C'(r)$.

(c) Determine the interval in which $C(r)$ is decreasing.

(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 company models the profit from a new product launch using the function $P(t) = 3t(2 - e^{-t})$, where $P$ is the profit in thousands of dollars and $t$ is the time in months since launch.

Find $\frac{dP}{dt}$.

Find $\frac{d^2P}{dt^2}$.

The company observes that the rate of change of profit growth begins to slow down after a certain point, indicating a point of inflexion. Find the time, $t$, in months at which this point of inflexion occurs.

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$, and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$, for $0 \le x \le 40$.

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

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

Point A has coordinates $(0, 0)$, point B has coordinates $(0, 15)$, point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$.

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

(ii) Find the area of the park.

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

(ii) Calculate the percentage error in the landscaper's estimate.

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ($x=40$). Point Q lies on the northern boundary curve $g(x)$.

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

(ii) Find the largest area of the meditation garden.

A company's daily production cost, $C(x)$, in thousands of dollars, for producing $x$ units of a specialized component, is modelled by the function $C(x) = x^2 + \frac{54}{x}$, for $x > 0$.

Write down the equation of the vertical asymptote of $C(x)$.

Find $C'(x)$, the marginal cost function.

Determine the interval for $x$ where the production cost $C(x)$ is increasing.

(a) A confectionery company is designing a new wrapper for a chocolate bar. The wrapper is in the shape of a rectangular prism with a square base of side length $x$ cm. The height, $h$ cm, is twice the side length of the base.

Write down an expression for $h$ in terms of $x$.

(b) The chocolate bar has a volume of $250$ cm$^3$.

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

(c) Calculate the total external surface area of this rectangular prism wrapper.

(d) The company also considers a cylindrical wrapper with radius $r$ cm and height $H$ cm. This wrapper must also hold $250$ cm$^3$ of chocolate.

Find an expression for the height, $H$, of the cylindrical wrapper in terms of $r$.

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

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

(f) Find $\frac{dA}{dr}$.

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

(g.ii) Find the minimum value of $A$ for the cylindrical wrapper.

(h) To account for manufacturing waste and overlap, an additional $12\%$ of the calculated surface area is required for the rectangular prism wrapper, and $20\%$ for the cylindrical wrapper.

Determine which wrapper design the company should choose to minimize material usage. Justify your answer.

A spherical ice sculpture is melting in a gallery. Initially, the sculpture has a radius of 30 cm. This information is illustrated in the following diagram.

The gallery curator predicts that, as the ice sculpture melts, its radius will decrease at a constant rate of 0.5 cm per hour.

According to this model, find

(i) the radius of the ice sculpture, 10 hours after it begins melting.

(ii) the volume of the ice sculpture, 10 hours after it begins melting. Give your answer to one decimal place.

Let the function $V(t)$ represent the volume of the ice sculpture, $\text{cm}^3$, $t$ hours after it begins melting. $V(t)$ is given by

$V(t) = 113000 - 5000t + 150t^2 - 1.5t^3$, for $0 \le t \le 20$.

Find $V'(t)$.

Find the rate of change of the volume of the ice sculpture at $t = 10$ hours.

State one reason why the radius of the ice sculpture may not always decrease at a constant rate.

(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 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, $P(x)$.

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 $x = 2$.