Max / min points and solving f’(x)=0: notes and practice questions

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 group of engineers is designing a new observation Ferris wheel. The height, $H(t)$, in metres, of a passenger capsule above the ground is modelled by the function $H(t) = p \cos\left(\frac{\pi}{100}t\right) + q$, where $t$ is the time in seconds after the capsule begins its ascent from the highest point.

The lowest point a capsule reaches is 2 metres above the ground, and the highest point is 32 metres above the ground. The Ferris wheel completes one full rotation in 200 seconds.

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

Using your values from part (a), the function is $H(t) = 15 \cos\left(\frac{\pi}{100}t\right) + 17$.

(i) Find $H'(t)$.

(ii) Find $H''(t)$.

The engineers are particularly interested in the moment when the capsule's vertical speed is at its maximum, for the first time after $t=0$. This occurs at time $t=k$.

Calculate the value of $k$.

Calculate the height of the capsule at this time $k$.

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.

(a) The position of a reconnaissance drone, relative to a control tower, is given by the vector equation $r = \begin{pmatrix} 7 \\ 2 \end{pmatrix} + t \begin{pmatrix} -3 \\ 4 \end{pmatrix}$, where $r$ is the position vector in metres and $t$ is the time in minutes.

Write down the position vector of the drone when $t = 0$ and when $t = 1$.

(b) Calculate the speed of the drone.

(c) Find an expression for the distance of the drone from the origin at time $t$.

(d) Hence find the minimum distance of the drone from the origin and the time at which it occurs.

A landscape architect is designing a modular planter box for a new urban garden. The planter box has a base and top that are identical sectors of a circle, each with radius $r$ cm and angle $\theta$ radians. The height of the planter box is $h = 2$ cm. The total length of metal frame used for all edges of the planter box is $L = 20$ cm. This includes two circular arcs, four radial edges for the top and bottom sectors, and three vertical connecting edges.

(a) Show that $r = \frac{7}{2+\theta}$.

(b) The planter box is designed to hold soil, enclosing a volume, $V$.

(i) Find an expression for $V$ in terms of $\theta$.

(ii) Find the expression for $\frac{dV}{d\theta}$.

(iii) Solve algebraically $\frac{dV}{d\theta} = 0$ to find the value of $\theta$ that will maximize the volume, $V$.

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.

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.

The concentration of a certain chemical, $C$, in a solution over a period of time can be modelled using the function $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$, where $t$ is the time in hours after the experiment begins.

Sketch the graph of $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$ for $0 \le t \le 20$.

Find the concentration after $2$ hours.

Find the concentration after $15$ hours.

Find the maximum concentration and the time in hours at which this occurs.

Find the minimum concentration and the time in hours at which this occurs.

Find the times in hours when the concentration is $6$ mol/L.

A slope field for the differential equation $\frac{dy}{dx} = y - x^2$ is shown.

Some of the solutions to the differential equation have a local maximum point and a local minimum point.

Write down the equation of the curve on which all these maximum and minimum points lie.

Sketch this curve on the slope field.

The solution to the differential equation that passes through the point (0, 1) has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through (0, 1).

A drone launches a package, and its trajectory is modelled by the equation $h(x) = -0.015x^2 + 0.6x + 5$, where $h(x)$ is the height of the package in metres and $x$ is the horizontal distance in metres from the launch point.

On paper, sketch the graph of the path that the package flies for $x \ge 0$. Clearly indicate the initial height, the maximum height, and the horizontal distance when it lands.

Find the height of the package when it has travelled a horizontal distance of $15$ metres.

Find the maximum height of the package.

Find the horizontal distance at which the package lands on the ground, and explain what this value represents in the context of the problem.

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.

The number of visitors (in hundreds) to a new eco-tourism resort $t$ months after its opening is modelled by the function $N(t) = -0.001 t^3 + 0.045 t^2 - 0.375 t + 10$.

Sketch the graph of $N$ against $t$ for the first $40$ months, clearly indicating any intercepts and local extrema within this domain.

Find the maximum number of visitors (to the nearest whole number) during the first $40$ months.

Find the time(s) when the number of visitors is above $1200$. Give your answer in months, correct to two decimal places.

The relationship between the sound intensity, S, of a speaker and the distance, d, from the speaker can be modelled by $S = \frac{k}{d^2}$.

A sound engineer measures the sound intensity at different distances. The data collected is shown in the table.

The engineer finds the sum of square residuals in the form $1.0641k^2 - 209.36k + c$.

Find the exact value of c.

Hence find the least squares regression curve of the form $S = \frac{k}{d^2}$.

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.

(a) A drone's vertical velocity, $v$ metres per second, at time $t$ seconds, is given by $v = t \sin(2t^2)$.

Find an expression for the vertical acceleration of the drone.

(b) Hence, or otherwise, find its greatest vertical acceleration for $0 \le t \le 3$ seconds.

(c) The drone starts at ground level (displacement is 0). Find an expression for the vertical displacement of the drone.

(d) Hence show that the drone never descends below ground level.

(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 landscape architect is designing a series of modular planter boxes for an urban garden project. The volume, $V$ cm$^{3}$, of a particular planter box is modelled by the function $V(d) = 30d^2 - d^3$, where $d$ is the depth of the planter in cm.

(a) Use your graphic display calculator to find the value of $d$ that will produce the maximum volume.

(b) Show that the maximum volume of the planter box is $4000$ cm$^{3}$.

The width of the planter is $d$ cm, and the length is $(30 - d)$ cm.

(c) The architect is interested in the total linear dimension $L$, which is defined as the sum of the depth, width, and length of the planter. Hence find the value of $L$ when the volume is maximized.

(a) When the profit is zero, find the possible number of units produced, $x$.

(b) Determine the positive values of profit, $P$, for which there is only one positive value of $x$ (units produced).

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 $0 \le t \le 4$.