Second derivative (+max/min distinguishing): notes and practice questions

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 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 $k$.

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) 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.

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) 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 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) 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).

(a) A builder is designing a rectangular enclosure for a new community garden. The total length of fencing available for the enclosure is $60$ metres.

The builder wants to maximize the area of the garden. Calculate the maximum possible area of the enclosure.

(b) To ensure structural stability, the builder also needs to consider the sum of the squares of the lengths of the sides. Calculate the minimum possible value for the sum of the squares of the lengths of the sides of the enclosure.

(c) Justify that the value found in part (b) is indeed a minimum.

(a) The profit, $P$, in thousands of dollars, for a company producing $x$ hundred units of a certain product is modelled by the function $P(x) = -2x^2 + 20x - 15$, for $x \ge 0$.

Find an expression for $\frac{dP}{dx}$, the rate of change of profit with respect to the number of units produced.

(b) Hence, find the maximum profit the company can achieve.

(c) Justify that the profit you found in part (b) is a maximum point.

A civil engineer is designing a cross-section for a new pedestrian tunnel. The shape of the tunnel's ceiling can be modelled by the curve $y = 10 - \frac{x^2}{4}$, where $x$ and $y$ are measured in metres. The tunnel's base is along the $x$-axis, and it starts from the $y$-axis, extending to where the ceiling meets the $x$-axis.

(a) Sketch the curve, showing the region representing the cross-sectional area of the tunnel. Shade this region.

(b) Using the trapezium rule with five strips, determine an approximation for the cross-sectional area of the tunnel.

(c) Explain why your answer to part (b) will be an underestimate for the actual cross-sectional area.

(d) Using integration, determine the exact value of the cross-sectional area of the tunnel.

(e) Find the percentage error of your approximation from part (b) compared with the exact value found in part (d).

(a) The height of a drone above the ground, in metres, can be modelled by the function $H(t) = 0.5t^2 - 10t + 70$, where $t$ is the time in seconds after launch.

Find $H'(t)$.

(b) Find $H'(12)$ and $H'(32)$ and comment on the meaning of these values in the context of the drone's flight.

(c) Find the time when the drone reaches its minimum height and explain how you know that it is a minimum.

The diagram below shows a cross-section of a sculpted garden feature. The shaded area is bounded by the $x$-axis, the vertical line $x = 6$, the line $y = x-1$ and the curve $y = \frac{18}{x^2}$.

Using technology or otherwise, find the coordinates of points $P$, $Q$, $R$ and $S$.

Approximate the area of the cross-section using the trapezoidal rule with 5 strips of equal width over the interval $[1,6]$.

Explain why the approximation in part (b) is an overestimate.

Determine the exact area of the cross-section using integration.

Find the percentage error of your approximation from part (b), compared with the exact value from part (d).

The rate of a chemical reaction, $R$, in mol/s, at time $t$ seconds, is modelled by the function $R(t) = \frac{10}{t^2 - 3t + 5}$, for $t \ge 0$.

(a) Find an expression for $R'(t)$.

(b) Hence or otherwise, find the values of $t$ at which the rate of reaction has points of inflexion. Give your answers correct to three significant figures.