DP Mathematics: Analysis and Approaches Questionbank

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

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

Use an algebraic method to determine whether $f$ is a self-inverse function.

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.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

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

Hence find the area of the shaded region.

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

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

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

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 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 $f\left(0\right)$.

$x$-axis.

$y$-axis.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

Write down the equation of the vertical asymptote of the graph of $f$.

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

The oblique asymptote of the graph of $f$ can be written as $y=ax+b$ where $a, b\in \mathrm{\mathbb{Q}}$.

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

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

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

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 $m$.

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.

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

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

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

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 $c$.

Find the x-coordinate of P.

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

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 $k$.

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

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

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

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

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

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

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

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

Find the area of the region enclosed by the graphs of f and g.

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

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.

Write down the value of $q$;

Write down the value of $h$;

Write down the value of $k$.

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

Hence, find the area of the region enclosed by the graphs of $h$ and $h^{-1}$.

Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y=x$. There is a point $\text{P}(a,b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where .

Find the initial velocity of P.

Find the maximum speed of P.

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

Find the acceleration of P when it changes direction.

Find the total distance travelled by P.

Note:     In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$, is given by $v(t)=\frac{2t^2-4t}{t^2-2t+2}$, for $0⩽t⩽5$. The following diagram shows the graph of $v$

There are $t$-intercepts at $(0,0)$ and $(2,0)$.

Find the maximum distance of the particle from A during the time $0⩽t⩽5$ and justify your answer.

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 $p$ and of $q$.

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

Find the value of $p$.

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.

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.

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

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

A second particle Q also moves along a straight line. Its velocity, $v_{\text{Q}}\text{ m}{\text{s}}^{-1}$ after $t$ seconds is given by $v_{\text{Q}}=\sqrt{t}$ for $0⩽t⩽8$. After $k$ seconds Q has travelled the same total distance as P.

Find $k$.

Find the initial velocity of $P$.

Find the value of $p$.

(i)     Find the value of $q$.

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

(i)     Find the total distance travelled by P between $t=1$ and $t=p$.

(ii)     Hence or otherwise, find the displacement of P from A when $t=p$.

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

Find the total distance travelled in the first 5 seconds.

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

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 initial displacement of particle A from point P.

Find the value of $t$ when particle A first reaches point P.

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

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

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

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

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

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

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

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

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

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

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

Calculate the area of $A$.

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

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

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

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

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

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 $\frac{\text{d}y}{\text{d}x}$.

Find ${\int }_0^1\text{arccos}\left(\frac{x}{2}\right)\text{d}x$.

Hence, or otherwise, find $\underset{0}{\overset{\pi }{\int }}{\text{e}}^{-x}\text{sin}x\text{d}x$.

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

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

Find the $x$-coordinates of A and of C , giving your answers in the form $a+\text{arctan}b$, where $a$, $b\in \mathbb{R}$.

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

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.

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

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

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.

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

Find $p_{av}$(0.007).

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

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 $d$.

Show that $\frac{3}{x+1}-\frac{1}{x-1}=\frac{2x-4}{x^2-1}$.

The area enclosed by the graph of $y=f\left(x\right)$ and the line $y=4$ can be expressed as $\text{ln}v$. Find the value of $v$.

Given that ${\int }_0^{\text{ln}k}{\text{e}}^{2x}\text{d}x=12$, find the value of $k$.

Find the area of $R$.

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 $\frac{2\pi }{3}$.

Find an expression for the velocity, $v$ m s⁻¹, of the rocket during the first stage.

Find the distance that the rocket travels during the first stage.

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.