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

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

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

A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{\text{d}\theta }{\text{d}t}$, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

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

Given that $\underset{T\to \infty }{\lim }k=A$ and $\underset{T\to 0}{\lim }k=0$, sketch the graph of $k$ against $T$.

Determine the value of $a$ and of $b$. Give your answers correct to one decimal place.

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

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.

Express this volume in $\text{c}{\text{m}}^3$.

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 $d\in \mathbb{Z}$.

Find the point on the graph of $f$ at which the gradient of the tangent is equal to 6.

Calculate the volume, in cm³, of the bag.

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

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

Write down the coordinates of C, the midpoint of line segment AB.

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.

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

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

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.

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

Find the coordinates of P and Q.

Write down the derivative of $f$.

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

Find $f(2)$.

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

Find the x-coordinate of the point at which the normal to the graph of f has gradient $-\frac{1}{8}$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$.

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

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

Show that $a=8$.

Find f'(x)

Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

Find the equations of the tangents to this curve at the points where the curve intersects the line $x=1$.

Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel to the line $y=-x$.

Hence show that $f(x)=0$ can be expressed as $u^3-7u^2+15u-9=0$.

Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d are integers.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Sketch a graph of $y=f^{\prime }(x)$ for .

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 $y=mx+c$.

Given that the gradients of the tangents to C at P and Q are m₁ and m₂ respectively, show that m₁ × m₂ = 1.

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

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

Show that $\frac{\text{d}y}{\text{d}x}=-\left(\frac{1+y\text{sin}\left(xy\right)}{1+x\text{sin}\left(xy\right)}\right)$.

Determine an expression for $f^{\prime }(x)$ in terms of $x$.

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

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

Determine the values of $x$ for which $f(x)$ is a decreasing function.

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

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

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

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

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

Find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$.

Show that $k=6$.

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

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

Find $f^{\prime }\left(x\right)$.

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

Find the coordinates of the local minimum.

Find the gradient of the line DC.

Find the gradient of the graph of f at $x=-\frac{1}{2}$.

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

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

Determine the equation of the tangent to $C$ at the point $\left(\frac{2}{\text{e}},\text{ e}\right)$

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

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

Determine the equation of the normal at $x=-2$ 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 $y=g(x)$.

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

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

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

Find the value of this minimum area.

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

Express $\sin 2x$ in terms of $u$.

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

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

Interpret this point in the given context.