Sketch the curve $y=f\left(x\right)$ and the tangent to the curve at point $\text{A}$, clearly showing where the tangent crosses the $x$-axis.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

Find $f\left(4\right)$.

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

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.

Sketch the graph of the function g (x) = 10x + 40 on the same axes.

Write down the $y$-intercept of the graph of $y=f\left(x\right)$.

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

Find u.

The graph of $f$ has a horizontal tangent line at $x=0$ and at $x=a$. Find $a$.

Hence or otherwise, find an expression for the derivative of $f_n(x)$ with respect to $x$.

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

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

Sketch the graph of $y=f^{\prime }\left(x\right)$.

Hence find the equation of the normal to $C$ at the point (1, 1).

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

Write down the value of $f\prime \left(4\right)$.

By using mathematical induction, prove that

$f_n(x)=\frac{\sin 2^{n+1}x}{2^n\sin 2x},x\ne \frac{m\pi }{2}$ where $m\in \mathbb{Z}$.

Using implicit differentiation, find an expression for $\frac{\text{d}y}{\text{d}x}$.

Write down the coordinates of $\text{B}$.

Find $\left(f\circ f\right)\left(x\right)$.

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

Let $\text{C}$ be a point on the graph of $h$. The tangent to the graph of $h$ at $\text{C}$ is parallel to the graph of $f$.

Find the $x$-coordinate of $\text{C}$.

Sketch the graph of $y=f\left(x\right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.

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

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

Find the coordinates of P.

Find the equation of the tangent to the curve at the point $\left(\frac{1}{4}\text{, }\frac{5\pi }{6}\right)$.

Find the value of $\text{tan}\theta$.

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

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

Let $l$ be the tangent to the curve $y=x{\text{e}}^{2x}$ at the point (1, ${\text{e}}^2$).

Find the coordinates of the point where $l$ meets the $x$-axis.

Line $L$ passes through the origin and has a gradient of $\text{tan}\theta$. Find the equation of $L$.

Show that the line $y=x-1$ is tangent to the curve $y=f\left(x\right)$ at the point $\text{A}\left(4, 3\right)$.

Hence, or otherwise, prove that the tangent to the curve $y=g\left(x\right)$ at the point $\text{A}\left(a, g\left(a\right)\right)$ intersects the $x$-axis at the point $\text{R}\left(r, 0\right)$.

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

Find the equation of $L$.

Hence explain why the graph of $f$ has a local maximum point at $x=a$.

Write down the gradient of this tangent.

Find $f^{″}\left(b\right)$.

At the point (1, 1) , show that $\frac{\text{d}y}{\text{d}x}=\frac{2+\pi }{2-\pi }$.

Hence, show that $k=\text{e}+\frac{{\text{e}}^2}{4}$.

Find $f\text{'}\left(x\right)$.

Show that $h=\frac{\text{e}+6}{2}$.

Hence, write down $f^{-1}\left(x\right)$.

Find the exact coordinates of $\text{Q}$.

Show that $\frac{\text{d}y}{\text{d}x}=\frac{y \cos \left(xy\right)}{2y-x \cos \left(xy\right)}$.

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

Differentiate from first principles the function $f\left(x\right)=3x^3-x$.

Find the gradient of $L$.

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

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 $p$.

Prove that, when $\frac{\text{d}y}{\text{d}x}=0 , y=\pm 1$.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

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

The tangent to $C$ at the point Ρ is parallel to the $y$-axis.

Find the $x$-coordinate of Ρ.

Given that the gradient of $L$ is $\frac{1}{3}$, find the $x$-coordinate of $\text{B}$.

The normal to the graph of $f$ at $x=a$ and the tangent to the graph of $f$ at $x=b$ intersect at the point ($p$, $q$) .

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

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

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

Hence find the equation of the tangent to $C$ at the point where $x=1$.

Consider the curve with equation $y=(2x-1){\text{e}}^{kx}$, where $x\in \mathrm{ℝ}$ and $k\in \mathrm{\mathbb{Q}}$.

The tangent to the curve at the point where $x=1$ is parallel to the line $y=5{\text{e}}^{k}x$.

Find the value of $k$.

Find the equation of the tangent to $C$ at $\text{P}$.

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

Draw the line $N$ on the diagram above.

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

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

The line $y=kx-5$ is a tangent to the curve of $f$. Find the values of $k$.

Determine whether $f_n$ is an odd or even function, justifying your answer.

Find $h\left(4\right)$.

Hence, use your answer to part (d)(i) to show that the graph of $f$ has a local minimum point at $x=b$.

Find the acute angle between $y=x$ and $L$.

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

Hence find the coordinates of all points on $C$, for $0<x<4\mathrm{\pi }$, where $\frac{\text{d}y}{\text{d}x}=0$.

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

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

Show that, for $n>1$, the equation of the tangent to the curve $y=f_n(x)$ at $x=\frac{\pi }{4}$ is $4x-2y-\pi =0$.

Show that the equation of $L$ is $y=-x+2 \ln 45+2$.

Show that $\frac{dy}{dx}=\frac{3x^2}{2{\text{e}}^{2y}-1}$.

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

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

Find the value of $k$.

Find the gradient of the line DC.

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

Show that $a=8$.

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

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

Determine the range of $f\left(x\right)$ for $p$ ≤ $x$ ≤ $q$.

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

Hence find the equation of the tangent to the graph of $h$ at $x=4$.

Show that $\frac{dy}{dx}+\left(x\frac{{\displaystyle dy}}{{\displaystyle dx}}+y\right)\left(1+\ln \left(xy\right)\right)=1$.

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 $f(2)$.

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

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