7.3 Workshop Exercises 3

For more worked examples, see Gilbert and Jordan Guide to Mathematical Methods, pages 153–164.

True or False?

i) If a point $(x,y)\neq(0,0)$ has polar coordinates $(r,\theta)$ then $x^{3}+y^{3}=r^{3}(\cos^{3}\theta+\sin^{3}\theta)$.

ii) The set of points given in polar coordinates by $\theta=\frac{\pi}{4}$ is equal to the line $y=x$.

iii) The tangent line to the plane curve $f(x,y)=0$ at a point $P$ has gradient $\frac{f_{x}(P)}{f_{y}(P)}$.

iv) If $P$ is a stationary point of the function $f(x,y)$ of two variables and $f_{xx}>0$ at $P$, then $P$ is a local minimum.

The main themes this week are: parametrized curves and arc length (W3.1-3.4); Chain Rule 1 and implicit differentiation (W3.5-3.7); and stationary points for functions of two variables (W3.8-3.10).

W3.1. Let $x=\cosh t$ and $y=\sinh t$.

(i) Show that $(x,y)$ lies on the right branch of the hyperbola $x^{2}-y^{2}=1.$

(ii) Obtain expressions for $dx/dt$ and $dy/dt$, and hence $dy/dx$.

W3.2. Let $a>0$ and consider the parabola $C$ given by $Y^{2}=4aX.$

(i) Show that $P=(at^{2},2at)$ lies on the parabola.

(ii) Find the equation of the tangent to $C$ at $P$ and the normal to $C$ at $P$.

(iii)* Find the arc length along the parabola from $(0,0)$ to $(a,2a)$ (i.e. from $t=0$ to $t=1$).

W3.3. Let $T$ be the curve that is given by $y^{2}=x^{2}(x+1).$

(i) Show that $x=-\cos^{2}\theta$ and $y=-\sin\theta\cos^{2}\theta$ give a point on $T$, and obtain an expression for $dy/dx$ in terms of $\theta$.

(ii) Show that $x=\tan^{2}\theta$ and $y=\sec\theta\tan^{2}\theta$ also give a point on $T$, and obtain an expression for $dy/dx$ in terms of $\theta$.

W3.4. (i) Show that $x=t^{2}$ and $y=t^{3}$ with $0\leq t\leq a$ gives a point on the curve

in the first quadrant.

(ii) Calculate the arclength $L_{a}$ of this portion of the curve. (This essentially repeats a result from the notes, but by a different route.)

(iii) For $a=1$, use a calculator to determine the precise value of $L_{a}$; and then use Simpson’s rule to approximate the value of the integral. Observe how close the answers are!

W3.5. The formula $4x^{2}+xy+6y^{2}=4$ defines an ellipse. By implicit differentiation, or otherwise, find an expression for $dy/dx.$

W3.6. (i) Suppose that $y$ is defined implicitly as a function of $x$ by $x+y=\tan^{-1}y$. Show that $1+y^{2}+y^{2}{{dy}\over{dx}}=0.$

(ii) Now suppose that $y$ is defined implicitly as a function of $x$ by

Show that

W3.7 Tschirnhausen’s cubic is the curve $C$ that is defined implicitly by $8y^{2}=x^{2}(x+1)$.

(i) Find the values of $y$ such that $x=1.$

(ii) Obtain an expression for $dy/dx$, and hence find the equations of the tangent lines to $C$ at the points such that $x=1$.

W3.8. (i) Find the first and second order partial derivaties of the function given by

(ii) Find the stationary points of $g$.

(iii) Classify the stationary points of $g$.

W3.9. Find the stationary points of:

(i) $f(x,y)=x+y^{3}-ye^{x}$,     (ii) $g(x,y)=xy^{3}+\frac{1}{x+y}$,     (iii) $h(x,y)=x^{3}-3xy^{4}+3y^{4}$.

Show that each of the functions has at least one saddle point. (For (iii), you are not asked to classify all of the stationary points.)

W3.10. Find the stationary points of the function $f(x,y)=x^{4}+y^{4}-4xy+4$, and determine which of them are local maxima, local minima or saddle points.