DP Mathematics: Applications and Interpretation 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 $f\left(x\right)$.

Find in terms of $a$ the value of $x$ at which the maximum occurs.

Hence find the value of $a$ for which $y$ has the smallest possible maximum value.

Find $f\text{'}\left(x\right)$.

Show that Jorge’s model satisfies the differential equation

$\frac{dN}{dt}=rN$

Find the expression $\frac{dV}{d\theta }$.

Solve algebraically $\frac{dV}{d\theta }=0$ to find the value of $\theta$ that will maximize the volume, $V$.

Show that $r=\frac{6}{2+\theta }$.

Find an expression for $V$ in terms of $\theta$.

Find the equation of the regression line of $h$ on $t$.

Suggest why Eva’s use of the linear regression equation in this way could be unreliable.

Find the equation of the least squares quadratic regression curve.

Find the value of $k$.

Hence, write down a suitable domain for Eva’s function $h\left(t\right)=p{t}^2+qt+r$.

Interpret the meaning of parameter $a$ in the context of the model.

By solving the differential equation $\frac{dh}{dt}=-R^2\sqrt{70 560h}$, show that the general solution is given by $h=17 640{\left(c-R^{2}t\right)}^2$, where $c\in \mathrm{ℝ}$.

Show that $\frac{dH}{dt}\approx 0.2514-0.009873t-0.1405\sqrt{H}$, where $0\le t\le T$.

Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$.

Use the general solution from part (d) and the initial condition $h\left(0\right)=3.2$ to predict the value of $T$.

Find this new height.

Show that $\frac{dh}{dt}=-R^2\sqrt{70 560h}$.

Find $\frac{dy}{dx}$.

Find an expression for $\frac{dW}{dv}$.

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5 {\text{km\hspace{0.05em}h}}^{-1} {\text{minute}}^{-1}$.

Find the rate of change of $W$ at this time.

A function $f$ is of the form $f\left(t\right)=p{\text{e}}^{q \cos \left(rt\right)}, p, q, r\in {\mathrm{ℝ}}^{+}$. Part of the graph of $f$ is shown.

The points $\text{A}$ and $\text{B}$ have coordinates $\text{A}(0, 6.5)$ and $\text{B}(5.2, 0.2)$, and lie on $f$.

The point $\text{A}$ is a local maximum and the point $\text{B}$ is a local minimum.

Find the value of $p$, of $q$ and of $r$.

Hence show that the snail reaches the point $(10, 2)$ before the water does.

When $\text{W}$ has an $x$-coordinate equal to $1$, find the horizontal component of the velocity of $\text{W}$.

Find the time taken for the snail to reach the point $(10, 2)$.

Find $\frac{dy}{dx}$.

Write down an expression for $E$ as a function of time.

Hence find $\frac{\text{d}E}{\text{d}t}$.

Hence or otherwise find the first time at which the kinetic energy is changing at a rate of 5 J s⁻¹.

Show that $\frac{dk}{dT}$ is always positive.

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 $(g^{-1}{)}^{\prime }(x)=0$.

Find $\frac{\text{d}y}{\text{d}\theta }$

Hence find the values of θ for which $\frac{\text{d}y}{\text{d}\theta }=2y$.

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 $h(1)$.

Find $h^{\prime }(8)$.

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

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

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 $L$.

Find the derivative of $f$.

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.

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.

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

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

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

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

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