DP Mathematics: Analysis and Approaches Questionbank

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

Find the least value of $n$.

Find $\int \frac{6x}{x^2+4}\text{d}x$.

Let $R$ be the region enclosed by the graph of $f$, the $x$-axis and the lines $x=1$ and $x=3$. The area of $R$ is $19.6$, correct to three significant figures.

Find $f\left(x\right)$.

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

Show that $V=\frac{2500\text{π}h}{3}-\frac{\text{π}{h}^3}{3}$.

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

Using your answer to part (c), find the value of $h$ for which $V$ is a maximum.

Find the maximum volume of the bird bath.

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.

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 $\mathrm{ORST}$.

Show that the maximum value of $m$ is $\frac{27}{32}\sqrt{\frac{2}{3}}$.

Use the differential equation $\frac{dy}{dx}=\frac{y^2+3xy+2x^2}{x^2}$ to show that the points of zero gradient on the curve lie on two straight lines of the form $y=mx$ where the values of $m$ are to be determined.

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.

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

a point of inflexion with zero gradient.

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

Find the value of $t$ when the rate of change of the population is at its maximum.

exactly one $x$-axis intercept.

exactly three $x$-axis intercepts.

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

exactly two $x$-axis intercepts.

one local maximum point and one local minimum point.

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

no points where the gradient is equal to zero.

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

a local minimum point for even values of $n$, where $n>1$ and $a\in {\mathrm{ℝ}}^{+}$.

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

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 $f\left(0\right)$.

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.

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.

Write down the coordinates of the two points of inflexion on the curve $y^2=x^3+1$.

Find the value of this $x$-coordinate, giving your answer in the form $x=\sqrt{\frac{p\sqrt{3}+q}{r}}$, where $p, q, r\in \mathrm{\mathbb{Z}}$.

Write down the value of $a$.

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

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 $b$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

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 $x=1.5$.

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 x-coordinate of P.

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

Find $f^{″}\left(b\right)$.

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

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

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 $q$.

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 $k$.

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

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

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

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

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{32\pi R^3}{81}$.

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.

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

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

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

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

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

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.

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

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

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

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 the value of the curvature of the graph of $f$ at the local maximum point.

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

Find the value of $p$.

Write down the coordinates of A.

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

Find the coordinates of B.

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

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 coordinates of A.

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

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

Hence, write $f\left(x\right)$ in the form $p\text{cos}\left(x+r\right)$.

Find the maximum speed of the ball.

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

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.

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 $k$.

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 $\pi$.

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 $a$.

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

Find the $x$-coordinates of the points of inflexion of the graph of $f$.

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

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

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

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

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 $b$.

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 the cost of producing 70 shirts.

Find the value of s.

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

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

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

Show that $a=8$.

Find $f(2)$.

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

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

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

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

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

Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

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

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

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

Calculate the volume, in cm³, of the bag.

Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.

Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.

Use your answers to parts (c) and (d) to show that

$A=3x^2+\frac{10368}{x}$.

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

Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.

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.

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

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

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

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 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.

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 the value of $p$.

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

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

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

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

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.

Sketch a graph of $y=f^{\prime }(x)$ for .

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

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

$y=f(x)$;

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

Solve the equation $f(x)=0$, giving your answers in the form $\arctan k$ where $k\in \mathbb{Z}$.

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

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

Express $\sin x$ in terms of $u$.

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

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

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

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

Find the $x$-coordinate(s) of the point(s) of inflexion of the graph of $y=f(x)$, labelling these clearly on the graph of $y=f^{\prime }(x)$.

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

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

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

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

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

Express $\sin 2x$ in terms of $u$.

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 volume, V cm³ , of the new trash can is given by

$V=110\pi r^3$.

Find the value of $x$ that maximizes the area of the plot.

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.

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.

Show that the width of the plot, in metres, is given by $100-x$.

Write down the height of the cylinder.

The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.

State whether the designer’s claim is correct. Justify your answer.

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

Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.

Find the total volume of the trash can.

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

Write down the area of the plot in terms of $x$.

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