DP Mathematics: Applications and Interpretation Questionbank

Hence, or otherwise, find the maximum possible volume of the box.

The box will contain spherical chocolates. The production manager assumes that they can calculate the exact number of chocolates in each box by dividing the volume of the box by the volume of a single chocolate and then rounding down to the nearest integer.

Explain why the production manager is incorrect.

Find in terms of $a$ the value of $x$ at which the maximum occurs.

Hence find the value of $a$ for which $y$ has the smallest possible maximum value.

Find $f\text{'}\left(x\right)$.

Find $\frac{dL}{dx}$

Show that $L=2x+\frac{300}{x}+22$

Use differentiation to show that $12a+4b+c=0$.

Find an expression for the total length of the ribbon $L$ in terms of $x$ and $y$.

Hence or otherwise find the minimum length of ribbon required.

Solve $\frac{dL}{dx}=0$

Hence find the maximum height of the tunnel.

Hence find the maximum height of the tunnel.

Find $S\prime \left(x\right)$.

Solve $S\text{'}\left(x\right)=0$.

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

Identify the $x$ value of the point where $|h\prime (x\left)\right|$ has its maximum value.

Interpret this point in the given context.

Find the expression $\frac{dV}{d\theta }$.

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

Juri starts at a height of $60$ metres and finishes at $\text{F}$, where $x=f$.

Sketch a possible diagram of the hill on the following pair of coordinate axes.

Show that $r=\frac{6}{2+\theta }$.

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

A function $f$ is of the form $f\left(t\right)=p{\text{e}}^{q \cos \left(rt\right)}, p, q, r\in {\mathrm{ℝ}}^{+}$. Part of the graph of $f$ is shown.

The points $\text{A}$ and $\text{B}$ have coordinates $\text{A}(0, 6.5)$ and $\text{B}(5.2, 0.2)$, and lie on $f$.

The point $\text{A}$ is a local maximum and the point $\text{B}$ is a local minimum.

Find the value of $p$, of $q$ and of $r$.

Find the value of $t$ at which the goat is eating grass at the greatest rate.

Find the value of $t$ at which the goat is eating grass at the greatest rate.

Calculate the surface area of the box in cm².

Find the number of boxes that should be sold each week to maximize the profit.

Find the least number of boxes which must be sold each week in order to make a profit.

Find $P\left(x\right)$.

Calculate the length AG.

Hence find the maximum value of $C$ in terms of $k$. Give your answer in the form $p{k}^3$, where $p$ is a constant.

Find the maximum height reached by the ball.

Show that the volume of the cone may be expressed by $V=\frac{\pi }{3}\left(2R{h}^2-h^3\right)$.

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{32\pi R^3}{81}$.

The graph of $y=f\left(x\right)$ has a local maximum at A. Find the coordinates of A.

Show that there is exactly one point of inflexion, B, on the graph of $y=f\left(x\right)$.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

Find the area of the window in terms of P and $r$.

Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum.

Show that in this case the height of the rectangle is equal to the radius of the semicircle.

Consider the curve $y=\frac{1}{1-x}+\frac{4}{x-4}$.

Find the x-coordinates of the points on the curve where the gradient is zero.

Show that there are exactly two points on the curve where the gradient is zero.

Find the equation of the normal to the curve at the point P.

The normal at P cuts the curve again at the point Q. Find the $x$-coordinate of Q.

The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.

Write down the $x$-coordinate of the point of inflexion on the graph of $y=f\left(x\right)$.

find the value of $f\left(1\right)$.

find the value of $f\left(4\right)$.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

Write down the gradient of the curve of $f$ at P.

Find the equation of the normal to the curve of $f$ at P.

Determine the concavity of the graph of $f$ when and justify your answer.

Find the value of $p$.

Write down the coordinates of A.

Write down the rate of change of $f$ at A.

Find the coordinates of B.

Find the the rate of change of $f$ at B.

Let $R$ be the region enclosed by the graph of $f$ , the $x$-axis, the line $x=b$ and the line $x=a$. The region $R$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Consider $f(x)=\log k(6x-3x^2)$, for , where $k>0$.

The equation $f(x)=2$ has exactly one solution. Find the value of $k$.

Express h in terms of r.

Show that $C=20\pi r^2+\frac{320\pi }{r}$.

Given that there is a minimum value for C, find this minimum value in terms of $\pi$.

Find the coordinates of P.

Find $f^{\prime }\left(x\right)$.

Hence, find the equation of L in terms of $a$.

The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$.

Find the volume of the container.

Find the value of $p$ and of $q$.

During the interval $p$ < $t$ < $q$, he volume of water in the container increases by $k$ m³. Find the value of $k$.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

(i)     Find the value of $c$.

(ii)     Show that $b=\frac{\pi }{6}$.

(iii)     Find the value of $a$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.

Sketch the graph of $f^{″}$ on the grid below:

Find the $x$-coordinates of the points of inflexion of the graph of $f$.

Hence find the values of $x$ for which the graph of $f$ is concave-down.

Find the population of fish at $t$ = 10.

Find the rate at which the population of fish is increasing at $t$ = 10.

Find the value of $t$ for which the population of fish is increasing most rapidly.

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

Find $g^{\prime }(x)$.

Show that $k=6$.

Find the equation of the tangent to the graph of $y=g(x)$ at $x=2$. Give your answer in the form $y=mx+c$.

Use your answer to part (a) and the value of $k$, to find the $x$-coordinates of the stationary points of the graph of $y=g(x)$.

Find $g^{\prime }(-1)$.

Hence justify that $g$ is decreasing at $x=-1$.

Find the $y$-coordinate of the local minimum.

Find the value of k.

Using your value of k , find f ′(x).

Use your answer to part (b) to show that the minimum value of f(x) is −22 .

Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.

Find the cost of producing 70 shirts.

Find the value of s.

Find the number of shirts produced when the cost of production is lowest.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

Find $\frac{\text{dy}}{\text{dx}}$.

Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.

Write down the $y$-intercept of the graph.

Find $f^{\prime }(x)$.

Show that $a=8$.

Find $f(2)$.

Write down the $x$-coordinates of these two points;

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

Write down the number of possible solutions to the equation $f(x)=5$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Write down a formula for $A$, the surface area to be coated.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Show that $k=-6$.

Find the coordinates of the local minimum.

Write down the interval where the gradient of the graph of $f\left(x\right)$ is negative.

Determine the equation of the normal at $x=-2$ in the form $y=mx+c$.

Calculate the area of cloth, in cm², needed to make Haruka’s bag.

Calculate the volume, in cm³, of the bag.

Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.

Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.

Use your answers to parts (c) and (d) to show that

$A=3x^2+\frac{10368}{x}$.

Find $\frac{\text{d}A}{\text{d}x}$.

Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.

The cloth used to make Nanako’s bag costs 4 Japanese Yen (JPY) per cm².

Find the cost of the cloth used to make Nanako’s bag.

Find the value of $P$ if no vases are sold.

Differentiate $P\left(x\right)$.

Find $f^{\prime }\left(x\right)$.

Find the gradient of the graph of $y=f\left(x\right)$ at $x=2$.

Find the equation of the tangent line to the graph of $y=f\left(x\right)$ at $x=2$. Give the equation in the form $ax+by+d=0$ where, $a$, $b$, and $d\in \mathbb{Z}$.

Write down an expression for $V$, the volume (cm³) of the speaker, in terms of $r$, $l$ and $\mathrm{\pi }$.

Write down an equation for the surface area of the speaker in terms of $r$, $l$ and $\mathrm{\pi }$.

Given the design constraint that $l=\frac{150-2\mathrm{\pi }{r}^2}{\mathrm{\pi }r}$, show that $V=150r-\frac{2\mathrm{\pi }{r}^3}{3}$.

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

Using your answer to part (d), show that $V$ is a maximum when $r$ is equal to $\sqrt{\frac{75}{\mathrm{\pi }}} \text{cm}$.

Find the length of the cylinder for which $V$ is a maximum.

Calculate the maximum value of $V$.

Use your answer to part (f) to identify the shape of the speaker with the best quality of sound.