DP Mathematics: Analysis and Approaches Questionbank

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

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.

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 $f\left(x\right)$.

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 $q\in {\mathrm{\mathbb{Z}}}^{+}$.

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

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

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

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

Show that $h=\frac{\text{e}+6}{2}$.

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

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

Find $f\text{'}\left(x\right)$.

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

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

Show that the point $\left(\frac{a}{2}, f_n\left(\frac{a}{2}\right)\right)$ on the graph of $y=f_n\left(x\right)$ is always above the horizontal axis.

Show that $f\text{'}\left(x\right)=\frac{2x}{\sqrt{x^2}\left(x^2+1\right)}$ for $x\in \mathrm{ℝ}, x\ne 0$.

State the three solutions to the equation ${f_n}^{\text{'}}\left(x\right)=0$.

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

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 $x=4$.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

$f\left(\text{i}u\right)$.

$g\left(\text{i}u\right)$.

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

By substituting $Y=\frac{dy}{dt}$, show that $Y=B{\text{e}}^{2t}$ where $B$ is a constant.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

By solving the differential equation $\frac{dy}{dt}=y$, show that $y=A{\text{e}}^t$ where $A$ is a constant.

Show that $\frac{dx}{dt}-x=-A{\text{e}}^t$.

Hence find $y$ as a function of $t$.

Show that $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.

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

Hence show that $x=-\frac{B}{2}{\text{e}}^{2t}+C$, where $C$ is a constant.

Find the two values for $\lambda$ that satisfy $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Let the two values found in part (c)(ii) be ${\lambda }_1$ and ${\lambda }_2$.

Verify that $y=F{\text{e}}^{{\lambda }_{1}t}+G{\text{e}}^{{\lambda }_{2}t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$.

By differentiating $\frac{dy}{dt}=-x+y$ with respect to $t$, show that $\frac{d^{2}y}{d{t}^2}=2\frac{{\displaystyle dy}}{{\displaystyle dt}}$.

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

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

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 $k$.

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

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 $b$.

Find the acceleration of the particle when it changes direction.

Find the total distance travelled by the particle.

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.

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 $A$.

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.

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.

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 $S$.

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

Show that a finite limit only exists for $k=\frac{\mathrm{\pi }}{4}$.

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

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 $\frac{\text{1}}{5}{\text{e}}^{\frac{3}{2}}$.

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

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

Hence show that $f_{n+1}(x)+f_{n-1}(x)=2x{f}_n\left(x\right)$, $n\in {\mathbb{Z}}^{+}$.

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

local maximum points;

Hence express $f_3(x)$ as a cubic polynomial.

Find the maximum distance of the particle from O.

local maximum points;

local minimum points;

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

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

local minimum points.

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

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

Explain why $f$ is an even function.

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

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

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

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

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

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

Find an expression for the volume of water $V({\text{m}}^3)$ in the trough in terms of $\theta$.

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

Find $\frac{\text{d}y}{\text{d}x}$.

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

An earth satellite moves in a path that can be described by the curve $72.5x^2+71.5y^2=1$ where $x=x(t)$ and $y=y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{\text{d}x}{\text{d}t}=7.75\times {10}^{-5}$ when $x=3.2\times {10}^{-3}$, find the possible values of $\frac{\text{d}y}{\text{d}t}$.

Give your answers in standard form.

(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 $h^{\prime }(\pi )=\frac{-21!}{2}{\pi }^2$.

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 $f^{\prime }\left(x\right)$.

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

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.