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

Find $f\left(4\right)$.

Find $h\left(4\right)$.

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

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

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

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 $x$.

Find the coordinates of $\text{B}$.

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

Show that $\cos \theta =\frac{3}{4}$.

Given that $\tan \theta >0$, find $\tan \theta$.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

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

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

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.

Using the given information, complete the following Venn diagram.

Find the number of surveyed students who did not like any of the three flavours.

A student is chosen at random from the surveyed students.

Find the probability that this student likes kiwi fruit smoothies given that they like mango smoothies.

Find the probability that both spins are yellow.

Find the probability that at least one of the spins is yellow.

Write down the probability that the second spin is yellow, given that the first spin is blue.

Find the value of $t$ when $P$ reaches its maximum velocity.

Show that the distance of $P$ from $\mathrm{O}$ at this time is $\frac{88}{27}$ metres.

Sketch a graph of $v$ against $t$, clearly showing any points of intersection with the axes.

Find the total distance travelled by $P$.

Complete the Venn diagram for these students.

One of the students who joined the sports club is chosen at random. Find the probability that this student joined both clubs.

Determine whether the events $S$ and $M$ are independent.

Expand and simplify ${(1-a)}^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that $1-3 \cos 2x+3 {\cos }^2 2x- {\cos }^3 2x=8 {\sin }^6 x$.

Show that ${\int }_0^{m}f\left(x\right)dx=\frac{32}{7}{\sin }^7 m$, where $m$ is a positive real constant.

It is given that ${\int }_m^{\frac{\mathrm{\pi }}{2}}f\left(x\right)dx=\frac{127}{28}$, where $0\le m\le \frac{\mathrm{\pi }}{2}$. Find the value of $m$.

State the number of boys who answered questions in Portuguese.

Find the probability that the boy answered questions in Hindi.

Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.

Write down an expression for $y$ in terms of $x$.

Find an expression for $V$ in terms of $x$.

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

Find the value of $x$ for which $V$ is a maximum.

Justify your answer.

Find the maximum volume.

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

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

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

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

Find $f^{″}\left(b\right)$.

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

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 $q$.

Write down the equation of the axis of symmetry.

Write down the values of $h$ and $k$.

Point $\text{Q}$ has coordinates $(5, 12)$. Find the value of $a$.

Find the equation of $L$.

Find the values of $d$ for which $g$ is an increasing function.

Find the values of $x$ for which the graph of $g$ is concave-up.

Find the value of $q$.

Write down the value of the discriminant of $f\prime$.

Hence or otherwise, find the value of $p$.

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

Sketch the graph of $f″$, the second derivative of $f$. Indicate clearly the $x$-intercept and the $y$-intercept.

Write down the value of $a$.

Find the values of $x$ for which the graph of $f$ is concave-down. Justify your answer.

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 $k$.

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.

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

Find the coordinates of P.

Find all the values of $x$ where the graph of $f$ is increasing. Justify your answer.

Find the value of $x$ where the graph of $f$ has a local maximum.

Find the value of $x$ where the graph of $f$ has a local minimum. Justify your answer.

Find the values of $x$ where the graph of $f$ has points of inflexion. Justify your answer.

The total area of the region enclosed by the graph of $f\text{'}$, the derivative of $f$, and the $x$-axis is $20$.

Given that $f\left(p\right)+f\left(t\right)=4$, find the value of $f\left(0\right)$.

Find the area of the region enclosed by the graph of $f$ and the line $L$.

Find $\int {\left(f\left(x\right)\right)}^2\text{d}x$.

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.

Hence find $\int \left(3x^2+1\right)\sqrt{x^3+x}\text{d}x$.

Write down an expression for the area of $R$.

Hence find the exact area of $R$.

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 $a$.

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 $\pi$.

(i)     Find the first four derivatives of $f(x)$.

(ii)     Find $f^{(19)}(x)$.

(i)     Find the first three derivatives of $g(x)$.

(ii)     Given that $g^{(19)}(x)=\frac{k!}{(k-p)!}(x^{k-19})$, find $p$.

(i)     Find $h^{\prime }(x)$.

(ii)     Hence, show that $h^{\prime }(\pi )=\frac{-21!}{2}{\pi }^2$.

The expression $\frac{3\sqrt{x}-5}{\sqrt{x}}$ can be written as $3-5x^p$. Write down the value of $p$.

Hence, find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

Find an expression for the velocity of $P_1$ at time $t$.

Particle $P_2$ also moves in a straight line. The position of $P_2$ is given by $\mathit{r}=\left(\begin{array}{c}-1\\ 6\end{array}\right)+t\left(\begin{array}{c}4\\ -3\end{array}\right)$.

The speed of $P_1$ is greater than the speed of $P_2$ when $t>q$.

Find the value of $q$.

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

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

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

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 $p$.

Show that $P=-2x^2+2x+8$.

Find the dimensions of rectangle $\mathrm{ORST}$ that has maximum perimeter and determine the value of the maximum perimeter.

Find an expression for $A$ in terms of $x$.

Find the dimensions of rectangle $\mathrm{ORST}$ that has maximum area.

Determine the maximum area of rectangle $\mathrm{ORST}$.

Find the value of $a$.

Find the volume of the solid formed when the shaded region is revolved $360°$ about the $x$-axis.

The graph of a function $f$ passes through the point $\left(\ln 4, 20\right)$.

Given that $f\text{'}\left(x\right)=6{\text{e}}^{2x}$, find $f\left(x\right)$.

Find $f\text{'}\left(x\right)$.

Show that $h=\frac{\text{e}+6}{2}$.

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

Show that $\frac{dy}{dx}=\frac{1-4 \ln x}{x^5}$.

The graph of $f$ has a horizontal tangent at point $\text{P}$. Find the coordinates of $\text{P}$.

Given that $f\text{'}\text{'}\left(x\right)=\frac{20 \ln x-9}{x^6}$, show that $\text{P}$ is a local maximum point.

Solve $f\left(x\right)>0$ for $x>0$.

Sketch the graph of $f$, showing clearly the value of the $x$-intercept and the approximate position of point $\text{P}$.

Let $f^{\prime }(x)=\frac{3x^2}{{(x^3+1)}^5}$. Given that $f(0)=1$, find $f(x)$.

Find the value of $s\left(4\right)-s\left(2\right)$.

Find the total distance travelled in the first 5 seconds.

Let $f^{\prime }(x)={\sin }^3(2x)\cos (2x)$. Find $f(x)$, given that $f\left(\frac{\pi }{4}\right)=1$.

Find the equation of the axis of symmetry of the graph of $y=f(x)$.

Write down the value of $c$.

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

Let $f\left(x\right)=\frac{6-2x}{\sqrt{16+6x-x^2}}$. The following diagram shows part of the graph of $f$.

The region R is enclosed by the graph of $f$, the $x$-axis, and the $y$-axis. Find the area of R.

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

Write down the gradient of this tangent.

Find the value of $k$.

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 $k$.

Show that $f^{\prime }\left(x\right)=\frac{1-\text{ln}5x}{k{x}^2}$.

Find the x-coordinate of P.

Show that the x-coordinate of Q is $\frac{\text{1}}{5}{\text{e}}^{\frac{3}{2}}$.

The region R is enclosed by the graph of $f$, the x-axis, and the vertical lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of $k$.

Find $\int x{\text{e}}^{x^2-1}\text{d}x$.

Find $f(x)$, given that $f^{\prime }(x)=x{\text{e}}^{x^2-1}$ and $f(-1)=3$.

Write down an equation for the area, $A$, of the curved surface in terms of $r$.

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

Find the value of $r$ when the area of the curved surface is maximized.

Find the value of $p$.

Find the value of $a$.

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

Find the exact value of $a$.

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

Consider f(x), g(x) and h(x), for x∈$\mathbb{R}$ where h(x) = $\left(f\circ g\right)$(x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

Find the value of $p$.

Find the value of $q$.

Find the displacement of particle A from the origin when $t=q$.

Find the distance of particle A from the origin when $t=10$.

Find the value of $d$.

A second particle, particle B, travels along the same straight line such that its velocity is given by $v\left(t\right)=14-2t$, for $t\ge 0$.

When $t=k$, the distance travelled by particle B is equal to $d$.

Find the value of $k$.

Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.

Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.

Write down an expression for $W$ in terms of $p$.

Find the price, $p$, that will give Maria the highest weekly profit.

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

Find the gradient of the line DC.

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

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

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

Hence, find the number of vases that will maximize the profit.

Let $f(x)=15-x^2$, for $x\in \mathbb{R}$. The following diagram shows part of the graph of $f$ and the rectangle OABC, where A is on the negative $x$-axis, B is on the graph of $f$, and C is on the $y$-axis.

Find the $x$-coordinate of A that gives the maximum area of OABC.

The derivative of a function $f$ is given by $f^{\prime }(x)=2{\text{e}}^{-3x}$. The graph of $f$ passes through $\left(\frac{1}{3}\text{,}5\right)$.

Find $f(x)$.

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

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

Draw the line $N$ on the diagram above.

Find the $x$-coordinates of $\text{A}$ and $\text{B}$.

Show that the area of the shaded region is $12\mathrm{\pi }$.

Find the value of $l$.

Hence, find the volume of the cone.

Find the height of the ball above the ground when it is released.

Find the minimum height of the ball above the ground.

Show that the ball takes $2$ seconds to return to its initial height above the ground for the first time.

For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8+0.2\sqrt{2}$ metres above the ground.

Find the rate of change of the ball’s height above the ground when $t=\frac{1}{3}$. Give your answer in the form $p\mathrm{\pi }\sqrt{q} {\mathrm{ms}}^{-1}$ where $p\in \mathrm{\mathbb{Q}}$ and $q\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $\frac{dy}{dx}=\cos \left(x-\frac{\mathrm{\pi }}{4}\right)$ and $y=2$ when $x=\frac{3\mathrm{\pi }}{4}$, find $y$ in terms of $x$.

The derivative of a function $f$ is given by $f\text{'}\left(x\right)=3\sqrt{x}$.

Given that $f\left(1\right)=3$, find the value of $f\left(4\right)$.

Let $f^{\prime }\left(x\right)=\frac{8x}{\sqrt{2x^2+1}}$. Given that $f\left(0\right)=5$, find $f\left(x\right)$.

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