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

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

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

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

Find $f\text{'}\left(x\right)$.

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

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

Find $\frac{dy}{dx}$.

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

Find $\frac{dy}{dx}$.

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 $r$.

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

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, $V$.

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 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.

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

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

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

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

(i)     Find $h^{\prime }(x)$.

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

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

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.

Find the least value of $n$.

Explain why $f$ is an even function.

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

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

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

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

Find the derivative of $f$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find $h(1)$.

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

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

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

(ii)     Find $f^{(19)}(x)$.

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

Show that $r=\frac{6}{2+\theta }$.

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

Show that Jorge’s model satisfies the differential equation

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

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

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

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 $h\left(t\right)=p{t}^2+qt+r$.

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

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 $c\in \mathrm{ℝ}$.

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

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