5.14 Implicit differentiation and related rates questions for IB Maths AA

Find ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}$ in terms of $x$ and $y$.

Find the gradient of the curve at each of the points where $x=1$.

Show that $\theta =\arctan \left({\displaystyle \frac{3}{x}}\right)-\arctan \left({\displaystyle \frac{1.2}{x}}\right)$.

Find the maximum value of $\theta$ and the value of $x$ at which it occurs.

Find ${\displaystyle \frac{d\theta }{dx}}$.

Hence show that the maximum value of $\theta$ occurs when $x=\sqrt{3.6}$.

The visitor walks directly towards the wall at a constant speed of 0.5 m s${}^{-1}$.

Find the rate of change of $\theta$ when the visitor is 4 m from the wall.

Find the range of values of $x$ for which $\theta >0.3$.

By differentiating $\tan y=x$ implicitly with respect to $x$, show that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(\arctan x)={\displaystyle \frac{1}{1+x^2}}$.

Let $g(x)=\arctan x+\arctan \left({\displaystyle \frac{1}{x}}\right)$, for $x\mathrm{\ne }0$.

Show that $g^{\mathrm{\prime }}(x)=0$.

Hence find the value of $g(x)$ for $x>0$ and the value of $g(x)$ for $x<0$.

Explain why $g^{\mathrm{\prime }}(x)=0$ for all $x\mathrm{\ne }0$, but $g$ is not a constant function.

Let $a$ be a positive constant and let $h(x)=\arctan \left({\displaystyle \frac{a+x}{1-ax}}\right)-\arctan x$, for $x<{\displaystyle \frac{1}{a}}$.

Show that $h^{\mathrm{\prime }}(x)=0$.

Hence show that $\arctan a+\arctan x=\arctan \left({\displaystyle \frac{a+x}{1-ax}}\right)$ for all $x<{\displaystyle \frac{1}{a}}$.

Use part (c) to show that $\arctan \left({\displaystyle \frac{1}{2}}\right)+\arctan \left({\displaystyle \frac{1}{3}}\right)={\displaystyle \frac{\pi }{4}}$.

Use part (c) to show that, for integers $n\ge 2$,

Let $S_N={\displaystyle \sum _{n=2}^N\arctan \left(\frac{2}{n^2}\right)}$, where $N\ge 2$.

Hence show that $S_N={\displaystyle \frac{\pi }{4}}+\arctan \left({\displaystyle \frac{1}{2}}\right)-\arctan \left({\displaystyle \frac{1}{N}}\right)-\arctan \left({\displaystyle \frac{1}{N+1}}\right)$.

Hence show that ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\arctan \left(\frac{2}{n^2}\right)=\frac{3\pi }{4}}$.

Find the rate of change of the radius of the snowball when $r=4$.

Find the rate of change of the volume of the snowball at the same instant.

Find the speed at which P moves along the shoreline when $x=300$.

Find the distance OP at the instant when P is moving along the shoreline at $1000$ m s${}^{-1}$.

A ladder of length $5$ m slides so that its foot moves away from the wall at a constant speed of $0.3$ m s${}^{-1}$.

Find the speed at which the top of the ladder moves down the wall when the foot is $3$ m from the wall.

Explain why this model is unrealistic as the top of the ladder approaches the ground.

Consider a ladder of length $L$ m. Let M be the midpoint of [AB].

Show that, for every position of the ladder, M lies on a circle, and state the centre and radius of this circle.

Let $\theta$ be the angle between the ladder and the ground, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that the line (AB) has equation $x\sin \theta +y\cos \theta =L\sin \theta \cos \theta$.

Consider the curve $E$ with equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=L^{\frac{2}{3}}$, for $x>0$, $y>0$.

Use implicit differentiation to show that ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}=-{\left({\displaystyle \frac{y}{x}}\right)}^{\frac{1}{3}}$.

The point P has coordinates $(L{\cos }^3\theta ,L{\sin }^3\theta )$, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that P lies on $E$.

Show that the tangent to $E$ at P is the line (AB) from part (c).

Explain briefly the significance of the result in part (f) for the sliding ladder.

Now consider the family of curves $x^n+y^n=1$, for $x>0$, $y>0$, where $n>0$. The tangent to such a curve at the point $(p,q)$ meets the $x$-axis at X and the $y$-axis at Y.

Show that the tangent at $(p,q)$ has equation $p^{n-1}x+q^{n-1}y=1$.

Hence show that ${\mathrm{X}\mathrm{Y}}^2=p^{2-2n}+q^{2-2n}$.

Show that when $n={\displaystyle \frac{2}{3}}$, the length XY is the same for every point $(p,q)$ on the curve.

By considering the points $({\displaystyle \frac{1}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}})$ and $({\displaystyle \frac{\sqrt{3}}{2}},{\displaystyle \frac{1}{2}})$, show that the length XY is not constant when $n=2$.