5.4 Tangents and normals questions for IB Maths AA

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

Find the coordinates of the two stationary points of the graph of $f$.

Find the equation of the tangent to the graph of $f$ at the point where $x=4$.

Find the angle of inclination of the road at the point where $x=50$, giving your answer in degrees.

Find the values of $x$ at which the angle of inclination of the road is ${10}^{\circ }$.

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 $n=2$.

Show that $f(1+h)=2+h-h^2$.

The point $\mathrm{Q}(1+h,f(1+h))$, where $h\mathrm{\ne }0$, also lies on the graph of $f$.

Show that the gradient of the chord $[\mathrm{P}\mathrm{Q}]$ is $1-h$.

Hence find the equation of the tangent at P.

Write down the equation of $L$.

Find the $x$-coordinate of P.

Find the area of the region enclosed by $L$ and the graph of $f$.

Find the acute angle between the tangent to the graph of $f$ at O and the $x$-axis.

The line $y=x\tan \theta$, where $\theta$ is acute, divides the region enclosed by the graph of $f$ and the $x$-axis into two parts of equal area.

Find the value of $\theta$.

Write down the value of ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }f(x)}$.

Find the value of $f^{\mathrm{\prime }}(1)$.

Let $g(x)=2f(x)+x$. Find the equation of the tangent to the graph of $g$ at the point where $x=1$.