Hence show that the equation of the tangent to the curve at the point $\left(0.16, 0.4\right)$ is $y=1.25x+0.2$.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

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.

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 .

Use the symmetry of the graph to show that the second solution is $x=4$.

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

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

Show that the normal to the curve at the point where $x=1$ is $2y-x+3=0$.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

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

Use your answer to part (a) to find the gradient of $L$.

Find the equation of the normal to the curve $y=f\left(x\right)$ at $\left(1, 2\right)$ in the form $ax+by+d=0$, where $a, b, d\in \mathrm{\mathbb{Z}}$.

Write down the value of $f(1)$.

Write down the coordinates of A.

Use your graphic display calculator to find the zero of f (x).

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 value of k.

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

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

Show that the graph of g has a gradient of 6 at P.

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

Find $(g\circ f)(x)$.

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

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

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

Draw the line $N$ on the diagram above.

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 $f\left(2\right)$.

Write down the value of $k$.

Find the coordinates of B.

Using your value of k , find f ′(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 $y$.

(i)     Find the value of $c$.

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

(iii)     Find the value of $a$.

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

Write down the equation for the axis of symmetry of the graph.

Write down the $x$-intercepts of the graph.

Use your graphic display calculator to find the coordinates of the local minimum point.

Draw the tangent $T$ on your graph.

Show that $a=8$.

Find f'(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 $y=-x$.

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 $\tan x=2x$.

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 $y=mx+c$.

Let L₂ be the tangent to the graph of g at P. L₁ intersects L₂ at the point Q.

Find the y-coordinate of Q.

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

On graph paper, draw the graph of $y=f\left(x\right)$ for  $-10\le x\le 4$  and  $-14\le y\le 14$. Use a scale of $1 \text{cm}$ to represent $1$ unit on the $x$-axis and $1 \text{cm}$ to represent $2$ units on the $y$-axis.

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

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.

Write down $f^{\prime }\left(2\right)$.

At the point where $x=2$, the gradient of the tangent to the curve is $0.5$.

Find the value of $a$.

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

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.

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

Write down the equation of $T$.

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

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

Write down the gradient of this tangent.

Write down, in the form $y=mx+c$, the equation of $L_2$.

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

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

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.

Find the equation of this tangent. Give your answer in the form y = mx + c.

Find the value of $p$.

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

Find the volume of the container.

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

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

Show that $k=6$.

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

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

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

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

Find $f^{\prime }\left(x\right)$.

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

Find the equation of $N$. Give your answer in the form $ax+by+d=0$ where $a$, $b$, $d\in \mathbb{Z}$.

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

Find the value of $k$.

Find the coordinates of the local minimum.

Find the gradient of the line DC.

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

Find the coordinates of P.

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

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

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

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

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

Find the value of $c$.

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

Given $f\left(a\right)=5.5$ and $f\text{'}\left(a\right)=-6$, state whether the function, $f$, is increasing or decreasing at $x=a$. Give a reason for your answer.

Find the value of $p$ and of $q$.

(i)     Find $w$.

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

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

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

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

Write down $f\prime \left(x\right)$.

Find the area of triangle $\text{AOB}$ in terms of $k$.

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

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

Find an expression for $\frac{dy}{dx}$.

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

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