DP Mathematics: Analysis and Approaches Questionbank

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.

Show that the $x$-coordinate of $\text{P}$ is $\frac{1}{3}\left(2a+r\right)$.

You are not required to demonstrate a change in concavity.

Write down the coordinates of the vertex of the graph of g.

Find the initial velocity of P.

Find the acceleration of P when it changes direction.

Show that $\frac{dy}{dx}=\pm \frac{3x^2+1}{2\sqrt{x^3+x}}$, for $x>0$.

Find the total distance travelled in the first 5 seconds.

Find the value of $t$ when particle A first changes direction.

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

Write down the value of $a$.

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

Hence, or otherwise, state the condition on $k\in \mathbb{R}$ such that all roots of the equation $P\left(z\right)=k$ are real.

Find the acceleration of the particle when it changes direction.

A second ornament is in the shape of a cuboid with a rectangular base of length $2x \text{cm}$, width $x \text{cm}$ and height $y \text{cm}$. The cuboid has the same volume as the pyramid.

The cuboid has a minimum surface area of $S {\text{cm}}^2$. Find the value of $S$.

Hence, determine the maximum volume of the container.

Find the rate of change of the height of the water when the container is filled to half its maximum volume.

Use the substitution $y=vx$ to show that $x\frac{dv}{dx}=v^2-v-2$.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

The following diagram shows the curve $\frac{x^2}{36}+\frac{{\left(y-4\right)}^2}{16}=1$, where $h\le y\le 4$.

The curve from point $\text{Q}$ to point $\text{B}$ is rotated $360°$ about the $y$-axis to form the interior surface of a bowl. The rectangle $\text{OPQR}$, of height $h \text{cm}$, is rotated $360°$ about the $y$-axis to form a solid base.

The bowl is assumed to have negligible thickness.

Given that the interior volume of the bowl is to be $285 {\text{cm}}^3$, determine the height of the base.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Show that $\frac{d^{2}P}{d{t}^2}=k^{2}P\left(1-\frac{P}{N}\right)\left(1-\frac{2P}{N}\right)$.

Hence show that the population of marsupials will increase at its maximum rate when $P=\frac{N}{2}$. Justify your answer.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt=\ln \frac{P}{P_0}\left(\frac{N-P_0}{N-P}\right)$.

Factorize $x^2+3x+2$.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=f(x)$;

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

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.

On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.

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.

Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

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

Sketch the graph of $y=x^4-6x^3-2x^2+58x-51$, stating clearly the coordinates of any maximum and minimum points and intersections with axes.

Find $f\left(4\right)$.

Use integration by parts to find $\int {\left(\text{ln}x\right)}^2\text{d}x$.

Solve the differential equation given that $y=-1$ when $x=1$. Give your answer in the form $y=f\left(x\right)$.

Solve the differential equation and show that a general solution is $x^2+y^2=cx$ where $c$ is a positive constant.

Hence, find the exact values of $I_3$ and $I_4$.

Find the coordinates of A.

For the graph of $f$, write down the amplitude.

Find the value of the acceleration of P at time t₁.

Hence write down a lower bound for $\sum _{n=4}^{\mathrm{\infty }}\frac{1}{n^3}$.

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

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.

Find the height of the cylinder, h , of the new trash can, in terms of r.

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.

State the alternative hypothesis.

Write down the χ² statistic.

Write down the associated p-value.

Find the value of x.

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

Copy the probability tree diagram and write down the relevant probabilities along the branches.

120 contestants attempted this game.

Find the expected number of contestants who fell into a trap while attempting to pass through a door in the third wall.

the expected frequency of female students who chose to take the Chinese class.

Find the value of $p$.

Find the cost of producing 70 shirts.

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

Find the total volume of the trash can.

Explain where the condition $n⩾2$ was used in your proof.

Write down the first value of $t$ at which P changes direction.

Find the total distance travelled by P, for $0⩽t⩽8$.

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

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\frac{1}{f(x)}$;

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

Write down the probability that Ayako avoids the trap in this wall.

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

Using your graphic display calculator, find the value of r which maximizes the value of V.

Determine the value of $h$.

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

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

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

Write down the range of $f$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Write down the maximum and minimum value of $v$.

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 .

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

Find u.

Complete the Venn diagram for these students.

Solve f '(x) = f "(x).

Using the given information, complete the following Venn diagram.

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

(i)     Find $h^{\prime }(x)$.

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

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

Write down $n\left(S\cap \left(M\cup B\right)\right)$.

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

Find the rate of change of $f$ at $\text{B}$.

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

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

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

Write down $n\left(B\cap M\cap S^{\prime }\right)$.

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

${\int }_{-2}^0\left(f\left(x\right)\text{ + 2}\right)\text{d}x$.

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

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

Write down the coordinates of A.

Find the coordinates of B.

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.

Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Find $\frac{\text{d}y}{\text{d}x}$.

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

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

Find the total distance travelled by P.

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

Show that $t$ + 1 is an integrating factor for this differential equation.

local minimum points;

Find the maximum distance of the particle from O.

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

Given that $y\left(1\right)=1$, use Euler’s method with step length $h$ = 0.25 to find an approximation for $y\left(2\right)$. Give your answer to two significant figures.

Find the value of y.

Show that the general solution of this differential equation is $P=A{\text{e}}^{kt}$, where $A\in \mathbb{R}$.

$\underset{t\to \mathrm{\infty }}{\text{lim}}P$

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

the solution of the differential equation, giving your answer in the form $P=f\left(t\right)$.

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

Find the value $\kappa$ for $x=\frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.

Explain why $f$ is an even function.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

Find ${\int }_{0.111}^{3.31}\left(h(x)-x\right)\text{d}x$.

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

Sketch the isoclines for $\frac{\text{d}y}{\text{d}x}=4$.

Find the equation of the tangent to the graph of $f$ at $\text{A}$.

Show that $P_i\left(n\right)=2n\text{sin}\left(\frac{\pi }{n}\right)$.

A function $f$ satisfies the conditions $f\left(0\right)=-4$, $f\left(1\right)=0$ and its second derivative is $f^{″}\left(x\right)=15\sqrt{x}+\frac{1}{{\left(x+1\right)}^2}$, $x$ ≥ 0.

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

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 $x$.

Show that $f^{\prime }\left(0\right)=0$.

Hence, show that $\int {\text{e}}^x\text{cos}{}^{2}x\text{d}x=\frac{{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{10}\text{cos}2x+\frac{{\text{e}}^x}{2}+c$,  $c\in \mathbb{R}$.

By writing $P_c\left(n\right)$ in the form $\frac{2\text{tan}\left(\frac{\pi }{n}\right)}{\frac{1}{n}}$, find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_c\left(n\right)$.

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

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

Find the value of $t$ at which the amount of salt in the tank is decreasing most rapidly.

Find the $x$-coordinates of the points of intersection of the two graphs.

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

Find the value of $p$.

Show that the function $f$ has a local maximum value when $x=\frac{3\pi }{4}$.

Find $f^{\prime }\left(x\right)$.

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 $m\in \mathbb{Z}$.

Find $\int f(x)\cos x\text{d}x$.

Sketch the graph of $y=f\left(x\right)$ for $1⩽x⩽1.4$ .

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

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

the value of $t_1$.

Find the total distance travelled by particle A in the first 3 seconds.

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

Find the perimeter of a regular hexagon, of side length, $x$ units, inscribed in a circle of radius 1 unit.

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

(ii)     Find $f^{(19)}(x)$.

Hence, or otherwise, solve the differential equation

$$\frac{\text{d}y}{\text{d}x}=\frac{x^2+3xy+y^2}{x^2},x>0,$$

given that $y=1$ when $x=1$. Give your answer in the form $y=g(x)$.

Find the probability that at least one of the two students is female.

the ${\chi }^2$ statistic.

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

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

Find when the particle is at rest.

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

Estimate the number of employees, from this 38, who are allergic to nuts.

Calculate the vertical distance Xavier travelled in the first 10 seconds.

Write down the null hypothesis, H₀ , for this test.

Find the volume of the container.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Consider the differential equation

$$\frac{\text{d}y}{\text{d}x}=f\left(\frac{y}{x}\right),x>0.$$

Use the substitution $y=vx$ to show that the general solution of this differential equation is

$$\int \frac{\text{d}v}{f(v)-v}=\ln x+\text{ Constant.}$$

Find the initial velocity of $P$.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Write down the value of b.

Find the number of employees who, in the last year, did not travel to work by car, bicycle or public transportation.

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 $k$.

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

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 an expression for the acceleration of P at time t.

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

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

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

Calculate the area of $A$.

By finding a suitable number of derivatives of $f$, find the first two non-zero terms in the Maclaurin series for $f$.

Find the acceleration of the particle at the instant it first changes direction.

Find the maximum speed of the ball.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

Find the coordinates of P.

Hence show that the Maclaurin series for $f\left(x\right)$ up to and including the term in $x^4$ is $x^2+\frac{1}{3}{x}^4$.

Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left(1+x\right)}^q\text{,}q\in \mathbb{Q}$.

Hence or otherwise find ${\int }_0^{\frac{\pi }{6}}\left(\text{sec}2x+\text{tan}2x\right)\text{d}x$ in the form $\text{ln}\left(a+\sqrt{b}\right)$ where $a$, $b\in \mathbb{Z}$.

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

local minimum points.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y=\left|\frac{1}{f(x)}\right|$.

Repeat this process to find the first four terms in a power series for ${\left(1+x\right)}^{-3}$.

Find the number of students in the school that study Mathematics in English.

By substituting $x=0$, find the value of $a_0$.

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

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 probability that the boy answered questions in Hindi.

Find the maximum speed of P.

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

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

Given that x_{k + 1} = x_k + a, find a.

The following diagram shows part of the graph of $f$.

The region enclosed by the graph of $f$, the $x$-axis and the lines $x=-p$ and $x=p$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Use the substitution $x=\frac{\pi }{2}-u$ to show that $J_n=I_n$.

Show that ${\left(\text{sin}x+\text{cos}x\right)}^2=1+\text{sin}2x$.

Determine the first time t₁ at which P has zero velocity.

Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.

By differentiating the above equation twice, show that

$$\left(1-x^2\right)f^{\left(4\right)}\left(x\right)-5x{f}^{\left(3\right)}\left(x\right)-4f^{″}\left(x\right)=0$$

where $f^{\left(3\right)}\left(x\right)$ and $f^{\left(4\right)}\left(x\right)$ denote the 3rd and 4th derivative of $f\left(x\right)$ respectively.

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 $p$.

Hence solve this differential equation. Give the answer in the form $y=f(x)$.

Given that the initial population is 1000, $L=10000$  and $m=0.003$, find the number of years it will take for the population to triple.

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

Find the value of $a$.

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.

(i)     Find the value of $q$.

(ii)     Hence, find the speed of P when $t=q$.

Show that $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, where $m\in \mathbb{R}$.

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

Illustrate graphically the inequality .

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

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

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

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

Hence, write down the maximum speed of the body.

the number of years it will take for the population to triple.

Find the probability that this adult is allergic to nuts and the liquid turns blue.

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

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.

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

Prove that there are two horizontal tangents to the general solution curve and state their equations, in terms of $c$.

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

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

Hence, find the exact values of $J_5$ and $J_6$

Use integration by parts to show that $\int {\text{e}}^x\text{cos}2x\text{d}x=\frac{2{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{5}\text{cos}2x+c$,  $c\in \mathbb{R}$.

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.

Using the substitution $u=\text{sin}x$, find $\int \frac{\text{co}{\text{s}}^{3}x\text{d}x}{\sqrt{\text{sin}x}}$.

Find the value of $p$.

On a new set of axes, sketch the graphs of $y=f_2(x)$ and $y=f_4(x)$ for −1 ≤ $x$ ≤ 1.

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

Show that, if $f^{\prime }\left(x\right)=0$, then $x=2-\sqrt{3}$.

Show that $g\text{'}\left(x\right)=2\left(x-r\right)\left(x-a\right)+x^2-2ax+a^2+b^2$.

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 $\text{R}\left(r, 0\right)$.

Use an appropriate Maclaurin series expansion to find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_i\left(n\right)$ and interpret this result geometrically.

Hence describe numerically the horizontal position of point $\text{P}$ relative to the horizontal positions of the points $\text{R}$ and $\text{A}$.

Hence deduce that the curve $y^2=x^3+x$has no local minimum or maximum points.

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

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 $m$.

Use this series approximation for $f\left(x\right)$ with $x=\frac{1}{2}$ to find an approximate value for ${\pi }^2$.

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.

A particle moves in a straight line such that at time $t$ seconds $(t⩾0)$, its velocity $v$, in $\text{m}{\text{s}}^{-1}$, is given by $v=10t{\text{e}}^{-2t}$. Find the exact distance travelled by the particle in the first half-second.

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

Given that particles A and B start at the same point, find the displacement function $s_{\text{B}}$ for particle B.

Find $\text{E}\left(X\right)$.

Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for $x⩾0$ and the $y$-axis for $y⩾0$ is rotated through $2\pi$ about the $x$-axis.

Hence, find an approximate value for ${\int }_0^1{\text{e}}^{x^2} \sin \left(x^2\right)dx$.

Show that $g\left(x\right)$ satisfies the equation $g\text{'}\text{'}\left(x\right)=2(g\text{'}(x)-g(x\left)\right)$.

Hence, deduce that $g^{\left(4\right)}\left(x\right)=2\left(g\text{'}\text{'}\text{'}\left(x\right)-g\text{'}\text{'}\left(x\right)\right)$.

Using the result from part (c), find the Maclaurin series for $g\left(x\right)$ up to and including the $x^4$ term.

Hence, or otherwise, determine the value of $\underset{x\to 0}{\lim }\frac{{\text{e}}^x \cos x-1-x}{x^3}$.

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

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

The function $f$ is defined by $f\left(x\right)={\left(x-1\right)}^2$, $x$ ≥ 1 and the function $g$ is defined by $g\left(x\right)=x^2+1$, $x$ ≥ 0.

The region $R$ is bounded by the curves $y=f\left(x\right)$, $y=g\left(x\right)$ and the lines $y=0$, $x=0$ and $y=9$ as shown on the following diagram.

The shape of a clay vase can be modelled by rotating the region $R$ through 360˚ about the $y$-axis.

Find the volume of clay used to make the vase.

Find the equation of $L$.

Find the displacement of the particle when $t=t_1$

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.

The range of $f$ is $a\le y\le b$, where $a, b\in \mathrm{ℝ}$.

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

By using the substitution $u=\text{sec} x$ or otherwise, find an expression for $\underset{0}{\overset{\frac{\pi }{3}}{\int }}{\text{sec}}^n x \tan x dx$ in terms of $n$, where $n$ is a non-zero real number.

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

Find the total distance travelled by the particle.

the distance travelled between $t=0$ and $t=t_1$.

Hence, find the area enclosed by the graph of $f$ and the graph of $f^{-1}$.

Show that the volume, $V {\text{m}}^3$, of water in the container when it is filled to a height of $h$ metres is given by $V=\pi \left(\frac{1}{3}{h}^3+h\right)$.

Find the time it takes to fill the container to its maximum volume.

Using the graph of $y=\frac{8x+x^4}{4-x^3}$, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of $y$ at $x=1.5$.

Find the times when the particle’s acceleration is $-1.9 {\text{m\hspace{0.05em}s}}^{-2}$.

Find the particle’s acceleration when its speed is at its greatest.

Show that $A=\frac{\sqrt{2}\mathrm{\pi }}{2}$.

Find the value of $k$.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Using l’Hôpital’s rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Let $D\left(t\right)$ represent the distance between airplane $A$ and airplane $B$ for $0\le t\le 2.5$.

Find the minimum value of $D\left(t\right)$.

Hence determine the maximum value of $\frac{dP}{dt}$ in terms of $k$ and $N$.

Find the value of $T$.

Find the coordinates of the points on the curve $y^3+3x{y}^2-x^3=27$ at which $\frac{\text{d}y}{\text{d}x}=0$.

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

Hence find the value of $\underset{0}{\overset{1}{\int }}\frac{1}{{\left(x^2+1\right)}^2}\text{d}x$.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

Show that, at these times, $\tan 6t=2$.

On the same set of axes, sketch and label isoclines for $\frac{\text{d}y}{\text{d}x}=-1, 0$ and $1$, and clearly indicate the value of each $y$-intercept.

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 the maximum displacement of $P$, in metres, from its initial position.

Hence find the area of the shaded region.

Hence or otherwise, explain why the point $(1, 1)$ is a local minimum.

Find the solution of the differential equation $\frac{\text{d}y}{\text{d}x}=x-y$, which passes through the point $(1, 1)$. Give your answer in the form $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$ on the same set of axes as part (a)(i).

Use your answer to part (a)(i) to write down an estimate for $f\left(0.4\right)$.

Find the value of $a$.

Show that $f\text{'}\text{'}\left(x\right)=\frac{24-6x^2}{{\left(x^2+4\right)}^2}$.

Write down the gradient of this tangent.

The curve has a point of inflexion at $\left(x_1, y_1\right)$ where ${\text{e}}^{-\frac{\mathrm{\pi }}{2}}<x_1<{\text{e}}^{\frac{\mathrm{\pi }}{2}}$. Determine the coordinates of this point of inflexion.

Find $f^{″}\left(b\right)$.

Find the exact values of $T_0$ and $T_1$.

Determine the coordinates of point $\text{A}$ and the coordinates of point $\text{B}$.

Write down the values of $t$ when $a=0$.

Find the value of $\underset{4}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x^3}\text{d}x$.

find the value of $f\left(1\right)$.

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

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

Expand ${\left(1+x\right)}^5$ using the Binomial Theorem.

Find his velocity when $t=15$.

Use Euler’s method, with step length $h=0.1$, to find an approximate value of $y$ when $x=1.4$.

By finding $g^{\prime }(x)$ explain why $g$ is an increasing function.

Consider a square of side length, $x$ units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.

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

Find the x-intercept of the graph of $f$.

Hence, find the area of the region enclosed by the graphs of $f$ and $g$.

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

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

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of $\pi$.

Determine the least value for $n$ such that the lower bound and upper bound approximations are both within 0.005 of $\pi$.

Use mathematical induction to prove that $f^{\left(n\right)}\left(x\right)={\left(-\frac{1}{4}\right)}^{n-1}\frac{\left(2n-3\right)!}{\left(n-2\right)!}{\left(1+x\right)}^{\frac{1}{2}-n}$ for $n\in \mathrm{\mathbb{Z}}, n\ge 2$.

Let $g\left(x\right)={\text{e}}^{mx}, m\in \mathrm{\mathbb{Q}}$.

Consider the function $h$ defined by $h\left(x\right)=f\left(x\right)\times g\left(x\right)$ for $x>-1$.

It is given that the $x^2$ term in the Maclaurin series for $h\left(x\right)$ has a coefficient of $\frac{7}{4}$.

Find the possible values of $m$.

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

Find the area of the region enclosed by the graph of $f$ and the $x$-axis.

The curvature at any point $(x,y)$ on a graph is defined as $\kappa =\frac{\left|\frac{{\text{d}}^{2}y}{\text{d}{x}^2}\right|}{{\left(1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^2\right)}^{\frac{3}{2}}}$.

Find $f\text{'}\left(x\right)$.

Show that $h=\frac{\text{e}+6}{2}$.

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

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

Calculate $\frac{\text{d}\theta }{\text{d}t}$ when $\theta =\frac{\pi }{3}$.

Find the $x$-coordinate of the point where $L$ intersects the line $y=x$.

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

Hence, find the area of $A$.

Use an appropriate trigonometric identity to show that $f_{n+1}(x)=\text{cos}\left(n\text{arccos}x\right)\text{cos}\left(\text{arccos}x\right)-\text{sin}\left(n\text{arccos}x\right)\text{sin}\left(\text{arccos}x\right)$.

Find the $x$-coordinate of the point of inflexion.

Solve the equation ${f_n}^{\mathrm{\prime }}(x)=0$ and hence show that the stationary points on the graph of $y=f_n(x)$ occur at $x=\text{cos}\frac{k\pi }{n}$ where $k\in {\mathbb{Z}}^{+}$ and 0 < $k$ < $n$.

Find the acceleration of the particle when $t=7$.

Hence find the value of $\frac{1}{2}\underset{1}{\overset{9}{\int }}\frac{\text{d}x}{x^{\frac{3}{2}}+x^{\frac{1}{2}}}$, expressing your answer in the form arctan $q$, where $q\in \mathbb{Q}$.

Deduce a similar expression for $v(T+k)$ in terms of $k$.

Use an appropriate trigonometric identity to show that $f_2(x)=2x^2-1$.

Deduce the set of values for $p$ such that there are two points on the curve $y=f\left(x\right)$ where $\frac{\text{d}y}{\text{d}x}=0$. Give a reason for your answer.

the number of years it will take for the population to triple.

Hence find the exact area of $R$.

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

Find the initial displacement of particle A from point P.

Explain where the condition $n⩾2$ was used in your proof.

Using the substitution $x=\tan \theta$ show that $\underset{0}{\overset{1}{\int }}\frac{1}{{\left(x^2+1\right)}^2}\text{d}x=\underset{0}{\overset{\frac{\pi }{4}}{\int }}{\cos }^2\theta \text{d}\theta$.

Use L’Hôpital’s rule to determine the value of

$$\underset{x\to 0}{\text{lim}}\left(\frac{{{\text{e}}^{-3x}}^{{}^2}+3\text{cos}\left(2x\right)-4}{3x^2}\right)$$

Explain why the inverse function $f^{-1}$ does not exist.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Show that $\sqrt{x^2+1}$ is an integrating factor for this differential equation.

By using the substitution $x^2=2\sec \theta$, show that $\int \frac{\text{d}x}{x\sqrt{x^4-4}}=\frac{1}{4}\arccos \left(\frac{2}{x^2}\right)+c$.

Use l’Hôpital’s rule to determine the value of

$\underset{x\to 0}{\lim } \frac{2 \sin x-\sin 2x}{x^3}$.

Find an expression for $s$ in terms of $t$.

Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating $f(0.4)$, using the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

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 $q$.

The folium of Descartes is a curve defined by the equation $x^3+y^3-3xy=0$, shown in the following diagram.

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the $y$-axis.

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

Find the largest possible domain $D$ for $f$ to be a function.

Find the least value of $n$.

Write down an equation in $r$ and $h$ that shows this information.

On the same set of axes, sketch the graphs of $y=f_1(x)$ and $y=f_3(x)$ for −1 ≤ $x$ ≤ 1.

For the graph of $f$, write down the period.

Find the other value of $t$ when particles A and B meet.

The expression for $f\prime \left(x\right)$ can be written in the form $\frac{a}{x}+\frac{b}{k-x}$, where $a, b\in \mathrm{ℝ}$. Find $a$ and $b$ in terms of $k$.

Find $t_1$ and $t_2$.

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

State the cross-sectional radius of the bowl at this point.

Write down the first three terms of the binomial expansion of $(1+t{)}^{-1}$ in ascending powers of $t$.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

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

Find the percentage error when $y\left(2\right)$ is approximated by the final rounded value found in part (a). Give your answer to two significant figures.

Consider an equilateral triangle ABC of side length, $x$ units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.

The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of $\frac{2\pi }{3}$ at O, as shown in the following diagram.

Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to $3\sqrt{3}$ units.

Write down the value of $c$.

Show that ${f_n}^{\text{'}}\left(x\right)=n{x}^{n-1}\left(a-2x\right){\left(a-x\right)}^{n-1}$.

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 $k$.

Solve the differential equation, for , giving your answer in the form $y=f\left(x\right)$.

Write down an expression for $f\text{'}\left(x\right)$.

a point of inflexion with zero gradient.

no points where the gradient is equal to zero.

one local maximum point and one local minimum point.

exactly one $x$-axis intercept.

Interpret your answer to part (e)(i) geometrically.

Use the model to find the volume of the barrel.

Use the Maclaurin series for $\ln (1+x)$ to write down the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

Hence, find the value of ${\int }_{\frac{1}{2}}^{\frac{3}{2}}\frac{1}{\sqrt{2x-x^2}}\text{d}x$.

Find the value of $p$ and of $q$.

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

the population after 10 years

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.

To prevent leaks, a sealant is applied to the interior surface of the bird bath.

Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.

Use l’Hôpital’s rule to find

$\underset{x\to 1}{\lim }\frac{\cos \left(x^2-1\right)-1}{{\text{e}}^{x-1}-x}$.

Determine the values of $p$, $q$ and $r$.

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$-coordinate of A on the graph of $g$.

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

The region $R$ is now rotated about the $y$-axis, through $2\pi$ radians, to form a solid.

By writing $\text{si}{\text{n}}^{3}y$ as $\left(1-\text{co}{\text{s}}^{2}y\right)\text{sin}y$, show that the volume of the solid formed is $\frac{2\pi }{3}$.

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 $k$.

By solving an appropriate differential equation, show that the particle’s velocity at time $t$ is given by $v\left(t\right)=(1+v_0){\text{e}}^{-t}-1$.

Sketch the graph of $f$, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.

The region enclosed by the graph of $f$, the y-axis and the x-axis is rotated 360° about the x-axis.

Find the volume of the solid formed.

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

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

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

Find the value of $q$.

Show that $P_c\left(n\right)=2n\text{tan}\left(\frac{\pi }{n}\right)$.

Let $f(x)=(x^2+3{)}^7$. Find the term in $x^5$ in the expansion of the derivative, $f^{\prime }(x)$.

Hence find $\underset{x\to 0}{\text{lim}}\left(\frac{{\int }_0^x\left({{\text{e}}^{-3t}}^{{}^2}+3\text{cos}\left(2t\right)-4\right)\text{d}t}{{\int }_0^{x}3t^2\text{d}t}\right)$.

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

Find the exact area of the shaded region, giving your answer in the form $p\pi +q\sqrt{3}$, where $p$, $q\in \mathbb{Q}$.

The region enclosed by the graph of $f$, the $x$-axis and the $y$-axis is rotated 360º about the $x$-axis. Find the volume of the solid formed.

Find the x-coordinate of P.

Show that the $x$-coordinate(s) of the points on the curve $y=f\left(x\right)$ where $\frac{\text{d}y}{\text{d}x}=0$ satisfy the equation $x^{p-1}=\frac{1}{p}$.

Find the gradient of $L$.

Find the area enclosed by the curve and the $x$-axis between B and D, as shaded on the diagram.

Write down the number of times that the acceleration of P is 0 m s⁻² .

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

Hence, find the exact values of $T_2$ and $T_3$.

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

At the point on the curve where $x=4$, it is given that $\frac{\text{d}y}{\text{d}t}=-0.1 {\text{m\hspace{0.05em}s}}^{-1}$

Find the value of $\frac{\text{d}x}{\text{d}t}$ at this exact same instant.

the acceleration when $t=t_1$.

Hence find the first three non-zero terms of the Maclaurin series for $\frac{x}{1+x^2}$.

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

Express h in terms of r.

Find the maximum volume.

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

By using the result to part (b) (i), show that $v\left(T-k\right)={\text{e}}^k-1$.

Hence, using integration, find the power series for $\text{arctan}x$, giving the first four non-zero terms.

Justify your answer.

Find the total distance travelled by P when its velocity is increasing.

Solve the differential equation giving your answer in the form $y=f(x)$.

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

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 $p$.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

find the value of $f\left(4\right)$.

Find $p_{av}$(0.007).

Let P($a$, $b$) be the unique point where the curves $C_1$ and $C_2$ intersect.

Show that the tangent to $C_1$ at P is perpendicular to the tangent to $C_2$ at P.

Use the fact that $\text{ta}{\text{n}}^{2}x=\text{se}{\text{c}}^{2}x-1$ to show that $T_n=\frac{1}{n-1}-T_{n-2}\text{,}n⩾2$.

With reference to the curvature of your sketch in part (c)(iii), and without further calculation, explain whether you conjecture $f\left(1.4\right)$ will be less than, equal to, or greater than your answer in part (a).

Sketch the graph of $x$ versus $t$ for 0 ≤ $t$ ≤ 60 and hence find the maximum amount of salt in the tank and the value of $t$ at which this occurs.

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

Find the equation of the tangent to the curve $y={\text{e}}^{2x}–3x$ at the point where $x=0$.

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

Find the $x$-intercept of the graph of $f$.

Write down a definite integral to represent the area of $A$.

Show that $P=-2x^2+2x+8$.

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 $m=-\frac{6x}{{\left(x^2+2\right)}^2}$.

Find the inverse function $g^{-1}$ and state its domain.

Hence, or otherwise, show that ${f_n}^{\text{'}}\left(\frac{a}{4}\right)>0$, for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

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.

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

The empty barrel is being filled with water. The volume $V{\text{m}}^3$ of water in the barrel after $t$ minutes is given by $V=0.8(1-{\text{e}}^{-0.1t})$. How long will it take for the barrel to be half-full?

Show that $\text{sec}2x+\text{tan}2x=\frac{\text{cos}x+\text{sin}x}{\text{cos}x-\text{sin}x}$.

a point of inflexion with zero gradient for odd values of $n$, where $n>1$ and $a\in {\mathrm{ℝ}}^{+}$.

Consider the graph of $y=x^n{\left(a-x\right)}^n-k$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$, $a\in {\mathrm{ℝ}}^{+}$ and $k\in \mathrm{ℝ}$.

State the conditions on $n$ and $k$ such that the equation $x^n{\left(a-x\right)}^n=k$ has four solutions for $x$.

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

Find the $x$-coordinate of Ρ.

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Find the acceleration of the particle when $t=2$.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

Find the area, $A$, of the region enclosed by the two curves.

Use the substitution $y=vx$ to show that $\int \frac{dv}{f\left(v\right)-v}=\ln x+C$ where $C$ is an arbitrary constant.

Use l’Hôpital’s rule to determine the value of $\underset{x\to 0}{\lim }\left(\frac{2x \cos \left(x^2\right)}{5 \tan x}\right)$.

Use integration by parts to show that $I_n=\frac{n-1}{n}{I}_{n-2}\text{,}n⩾2$.

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

Find $\int \left(6x+7\right)dx$.

Hence, find an expression for $f\left(x\right)$.

the $y$-coordinate of the local maximum point is $2c^{\frac{3}{2}}+2$.

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

By solving an appropriate differential equation and using the result from part (b) (i), find an expression for $s_{\text{max}}$ in terms of $v_0$.

By using the expression for $f\text{'}\left(x\right)$ and the result $\sqrt{x^2}=\left|x\right|$, show that $f$ is decreasing for $x<0$.

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 ($p$, $q$) .