Differentiation (DP IB Applications & Interpretation (AI): SL): Exam Questions

The equation of a curve is $y=\frac{3}{2}{x}^2-15x+2$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

The gradient of the tangent to the curve at the point $\mathrm{A}$ is $-3$.

Find

(i) the coordinates of $\mathrm{A}$

(ii) the equation of the tangent to the curve at $\mathrm{A}$. Give your answer in the form $y=mx+c$.

Consider the function $f\left(x\right)=3x^7-12x$.

Find $f\text{'}\left(x\right)$.

Find the gradient of the graph of $f$ at $x=0$.

Find the coordinates of the points on the graph of $f$ where the normal has a gradient of $4$.

The equation of a curve is $y=4-\frac{4}{x}$.

Find the equation of the tangent to the curve at the point where $x=2$. Give your answer in the form $y=mx+c$.

Find the coordinates of the points on the curve where the gradient is $16$.

Consider the function $f\left(x\right)=\frac{4}{x}+\frac{2x^4}{5}-\frac{2}{5}$, $x\ne 0$.

Find

(i) $f\left(2\right)$

(ii) $f\text{'}\left(2\right)$.

The line $l$ is tangent to the graph of $y=f\left(x\right)$ at the point where $x=2$.

Find the equation of $l$. Give your answer in the form $y=mx+c$.

The graph of $y=f\left(x\right)$ and the line $l$ intersect again at the point $\mathrm{A}$.

Use your graphic display calculator to find the coordinates of $\mathrm{A}$.

Consider the function $f\left(x\right)=x^2-bx+c$, where $b,c\in \mathrm{ℝ}$.

Find $f\text{'}\left(x\right)$.

The tangent to the graph of $y=f\left(x\right)$ at $x=2$ has equation $y=x-1$.

Find the value of $b$.

Find the value of $c$.

The equation of the curve $C$ is $y=\frac{1}{35}{x}^5-\frac{3}{4}{x}^3+6x$. A section of the curve $C$ is shown on the diagram below.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

Points $A$ and $B$ represent the local maximums on the diagram above.

Write down the coordinates of

(i) $A$

(ii) $B$

There are two points, $\mathrm{R}$ and $\mathrm{S}$, along the curve $C$ at which the gradient of the normal to the curve $C$ is equal to $-\frac{1}{10}$.

Calculate the $x$-coordinates of points $\mathrm{R}$ and $\mathrm{S}$.

A company produces and sells cricket bats. The company's daily cost, $C$, in hundreds of Australian dollars (AUD), of producing $x$ hundred cricket bats in a day can be modelled by

$C\left(x\right)=6x^3-10x^2+10x+4$

Find the cost to the company on a day when no cricket bats are produced.

The company's daily revenue, $R$, in hundreds of AUD, from selling $x$ hundred cricket bats is given by $R\left(x\right)=42x$.

Given that profit = revenue - cost, find an expression for the daily profit, $P\left(x\right)$, in hundreds of AUD, from selling $x$ hundred cricket bats.

Find $P\text{'}\left(x\right)$.

The derivative of $P\left(x\right)$ gives the marginal profit. Production is at its profit-maximising level when the marginal profit is zero and $P\left(x\right)$ is positive.

Find the profit-maximising production level and the expected profit.

Dora makes an open container for collecting strawberries from a rectangular piece of cardboard measuring 55 cm by 28 cm. She cuts a square of side $x$ cm from each corner, as shown in the following diagram, and folds up the sides.

Show that the volume, $V$ cm³, of the container is given by

$V=4x^3-166x^2+1540x$

Find $\frac{\mathrm{d}V}{\mathrm{d}x}$.

Find

(i) the value of $x$ that maximises the volume of the container

(ii) the maximum volume of the container. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

The diagram below shows part of the graph of the function $y=f\left(x\right)$, where $f\left(x\right)=\frac{4}{x}+x^2-4$, $x>0$.

Calculate the instantaneous rate of change of $f\left(x\right)$ when $x=2$.

Calculate the average rate of change of $f\left(x\right)$ between $x=2$ and

(i) $x=3$

(ii) $x=2.5$

(iii) $x=2.25$

Explain what would happen if you continued to calculate the average rates of change in part (b), moving the second $x$ value closer and closer to 2 each time.

The equation of a curve is $y=x-\frac{9}{x}+8$, for $x>0$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

The gradient of the tangent to the curve at the point $\mathrm{A}$ is $2$.

Find the coordinates of $\mathrm{A}$.

Find the equation of the normal to the curve at $\mathrm{A}$. Give your answer in the form $ax+by+d=0$, where $a$, $b$ and $d$ are integers.

Let $f\left(x\right)=a{x}^2+2x+9$. The gradient of the tangent to the graph of $f$ at the point $\left(-2,b\right)$ is $14$.

Find the value of $a$ and the value of $b$.

A curve has equation $y=\frac{1}{6}{x}^3-\frac{3}{8}{x}^2-\frac{3}{2}x+4$.

Find the coordinates of the points on the curve where the gradient is $2$. You must show all your working, and give your answers as exact fractions.

Find the values of $x$ for which the curve is increasing.

Two numbers, $x$ and $y$, are such that $x>y$ and the difference between them is 7.

Find the minimum possible value of the product $xy$, and the values of $x$ and $y$ that give this minimum.

A manufacturing company produces tins that must have a capacity of 470 cm³. Each tin is a closed cylinder with a height of $h$ cm and a base radius of $r$ cm.

Show that the surface area of a tin, $A$ cm², including the two circular ends, is given by

$A=2\pi r^2+\frac{940}{r}$

Sketch the graph of $A$ against $r$, for $r>0$.

The company would like to minimise the amount of metal used to make the tins.

(i) Find the coordinates of the stationary point on the graph of $A$, and justify that it is a minimum point.

(ii) Hence write down the minimum possible surface area of a tin, and the base radius that gives this minimum.

A tin of chopped tomatoes on sale in the UK has a capacity of 470 cm³ and a base radius of 3.7 cm.

Find the percentage difference between the surface area of this tin and the minimum possible surface area of a tin with the same capacity.

Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session. The height of the point of the javelin above the ground, $h$ metres, $t$ seconds after it is released can be modelled by

$h\left(t\right)=1.75+20.2t-4.90t^2$

Find $h\text{'}\left(t\right)$.

(i) Find the coordinates of the stationary point of the graph of $h$.

(ii) Justify that this stationary point is a maximum.

Find the greatest vertical distance that the point of the javelin travels above the height from which it was released.

A newly commissioned attack submarine is carrying out a series of manoeuvres to test its propulsion and steering systems. Its vertical position, $h$ metres, relative to sea level ($h=0$), $t$ minutes after it begins the manoeuvres, is given by

$h\left(t\right)=0.0125t^3-1.03t^2+16.6t-165$, for $0\le t\le 60$.

Find the coordinates of the stationary points of the graph of $h$.

For each of the stationary points found in part (a), determine whether it is a maximum point or a minimum point. Justify your answers.

Explain why, to find the maximum and minimum depths reached by the submarine for $0\le t\le 60$, it is not enough to consider only the stationary points found in part (a).

Find the greatest vertical distances that the submarine travels above and below the depth at which it started its manoeuvres, for $0\le t\le 60$.

The volume of a sphere of radius $r$ is given by the formula $V=\frac{4}{3}\pi r^3$.

Find $\frac{\mathrm{d}V}{\mathrm{d}r}$.

Find the rate of change of $V$ with respect to $r$ when $r=5$. Give your answer in terms of $\pi$.

Show that $\frac{\mathrm{d}V}{\mathrm{d}r}$ is an increasing function for $r>0$.

A curve has equation $y=f\left(x\right)$, where $f\left(x\right)=\frac{1}{3}{x}^3-2x^2-4x+\frac{31}{3}$.

The points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is $1$, and the $x$-coordinate of $\mathrm{A}$ is negative.

Find the coordinates of $\mathrm{A}$ and $\mathrm{B}$.

Find the equation of

(i) the tangent to the curve at $\mathrm{A}$

(ii) the normal to the curve at $\mathrm{B}$.

The lines in part (b) meet at the point $\mathrm{C}$.

Find the coordinates of $\mathrm{C}$.

Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs. The company's annual profit, $P\left(x\right)$ thousand pounds (£), can be modelled by

$P\left(x\right)=0.32x^3-12.4x^2+150x-480$

where $x$ is the number of chess sets sold per year, in hundreds. Because of manufacturing constraints, the company can sell at most 2500 chess sets in a year.

(i) State why there is no need to consider values of $x$ greater than 25.

(ii) Sketch the graph of $P\left(x\right)$ for $0\le x\le 25$.

(i) Find the coordinates of the stationary points of the graph, and state the number of chess sets sold and the profit that correspond to each point.

(ii) Find the maximum profit that the company can make in a year, and the number of chess sets that it must sell to make that profit.

Calculate

(i) the average rate of change of $P\left(x\right)$ between $x=5$ and $x=6$

(ii) the rate of change of $P\left(x\right)$ at $x=5$.

In each case, include the units and explain the meaning of the value you find.

State the values of $x$ for which the rate of change of $P\left(x\right)$ is negative. Explain the meaning of this result.

Let $f\left(x\right)=a{x}^2+bx+c$. The tangents to the graph of $f$ at the points $\left(-3,d\right)$ and $\left(2,-1\right)$ have gradients $25$ and $-5$ respectively.

Find the values of $a$, $b$, $c$ and $d$.

The diagram below shows part of the graph of the function $y=f\left(x\right)$, where $f\left(x\right)=9-\frac{1}{18}{x}^3-\frac{6}{x}$, $x>0$.

Calculate the average rate of change of $f\left(x\right)$ between $x=3$ and

(i) $x=4$

(ii) $x=3.5$

(iii) $x=3.25$.

Explain what would happen to the values of the average rates of change in part (a) if you continued to calculate them, moving the second $x$ value closer and closer to $3$ each time.

An engineer is designing a right cone that is to be produced on a 3D printer. The cone has a base radius of $r$ cm and a height of $h$ cm. The radius may vary, but the height is always 7 cm more than the radius.

Write down an expression for $V$, the volume of the cone, in terms of $r$ only.

The rate of change of $V$ with respect to $r$ is $\frac{5\pi }{3}{\text{ cm}}^3$ per cm at a particular value of $r$.

Find this exact value of $r$.

Two numbers, $x$ and $y$, are such that $x>y$ and the difference between them is $k$, where $k$ is a positive constant.

Find the minimum possible value of $x^2+3y^2$, and the values of $x$ and $y$ that give this minimum. Give your answers in terms of $k$.

Justify that your answer to part (a) is a minimum value.

An artist is producing large pieces of sculpture for an art installation. Each piece is in the form of a cylinder with base radius $r$ metres, with a hemisphere of the same radius fitted exactly on top of the cylinder.

Every surface of each piece of sculpture, including the base, must be painted, so the artist wants a design for which, for any given volume, the total surface area is the minimum possible.

Show that for a piece of sculpture with volume $k\pi$ m³, the minimum surface area occurs when

$r=\sqrt[3]{\frac{3k}{5}}$

Find the minimum possible surface area for a piece of sculpture with volume $\frac{40}{3}\pi$ m³. Give your answer as an exact value.

A curve has equation $y=f\left(x\right)$, where $f\left(x\right)=2x^3+\frac{3}{x}-4$.

The points $\mathrm{A}$ and $\mathrm{B}$ are the two points on the curve where the gradient is $3$, and the $x$-coordinate of $\mathrm{A}$ is negative.

Find the coordinates of $\mathrm{A}$ and $\mathrm{B}$.

Find the equation of

(i) the tangent to the curve at $\mathrm{A}$

(ii) the normal to the curve at $\mathrm{B}$.

The lines in part (b) meet at the point $\mathrm{C}$.

Find the coordinates of $\mathrm{C}$, giving your answers as exact fractions.

Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia. The company's monthly profit is $P\left(x\right)$ thousand pounds (£), where $x$ is the number of cribbage boards sold in the month, in hundreds.

For sales of between 0 and 100 cribbage boards in a month, the profit can be modelled by

$P\left(x\right)=4.53x^2-8.51$

For sales of between 100 and 1000 cribbage boards in a month, the profit can be modelled by

$P\left(x\right)=0.02x^3-\frac{9}{x}+5$

Because of manufacturing constraints, the company can sell at most 1000 cribbage boards in a month.

(i) Show that both formulae give the same profit for sales of 100 cribbage boards in a month.

(ii) State the values of $x$ for which each formula is valid.

On the same set of axes, sketch the graphs of the two profit functions. Sketch each function only over the values of $x$ for which it is valid.

Show that the combined profit function sketched in part (b) is increasing for all valid values of $x$ greater than zero.

Considering only values of $x$ for which $P\left(x\right)>0$, find the value of $x$ for which the rate of change of $P\left(x\right)$ is a minimum. Give this minimum rate of change, and explain its meaning in context.