## Monthly Archives: November 2018

### On the value of time…essence of time management

Time is Life.

Money lost can come back, but time lost can never come back. Time is more valuable than money. (Bertrand Russell).

### Mathematics versus Physics

The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.

In my opinion, for example, Boole’s Laws of (Human) Thought.

### An observation about Sir Isaac Newton

Newton’s patience was limitless. Truth, he said much later, was the offspring of silence and meditation. And, he said, I keep the subject constantly before me and wait till the first dawnings open slowly, by little and little into a full and clear light.

### Applications of Derivatives: Training for IITJEE Maths: Part V

Question I:

Show that the equation of the tangent to the curve $x=a \frac{f(t)}{h(t)}$ and $y=a \frac{g(t)}{h(t)}$ can be represented in the form: $\left| \begin{array}{ccc} x & y & a \\ f(t) & g(t) & h(t) \\ f^{'}(t) & g^{'}(t) & h^{'}(t) \end{array} \right|=0$

Question 2:

Show that the derivative of the function $f(x) = x \sin{\frac{\pi}{x}}$, when $x>0$ and $f(x)=0$, when $x=0$ vanishes on an infinite set of points of the interval \$latex (0,1), Hint: Use Rolle’s theorem.

Question 3:

Prove that $\frac{1}{1+x}<\log (1+x) < x$ for $x>0$. Use Lagrange’s theorem.

Question 4:

Find the largest term in the sequence $a_{n}=\frac{n^{2}}{n^{3}+200}$. Hint: Consider the function $f(x)=\frac{x^{2}}{x^{3}+200}$ in the interval $[1,\infty)$.

Question 5:

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Question 6:

Find the polynomial $f(x)$ of degree 6, which satisfies $\lim_{x \rightarrow 0} (1+\frac{f(x)}{x^{3}})=e^{2}$ and has a local maximum at $x=1$ and local minimum at $x=0$ and 2.

Question 7:

For the circle $x^{2}+y^{2}=r^{2}$, find the value of r for which the area enclosed by the tangents drawn from the point $(6,8)$ to the circle and the chord of contact is maximum.

Question 8:

Suppose that f has a continuous derivative for all values of x and $f(0)=0$, with $|f^{'}(x)| <1$ for all x. Prove that $|f(x)| \leq |x|$.

Question 9:

Show that $(e^{x}-1)>(1+x)\log {1+x}$, if $x \in (0,\infty)$.

Question 10:

Let $-1 \leq p \leq 1$. Show that the equations $4x^{3}-3x-p=0$ has a unique root in the interval $[1/2,1]$ and identify it.

### Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

Question 1.

If the point on $y = x \tan {\alpha} - \frac{ax^{2}}{2u^{2}\cos^{2}{\alpha}}$, where $\alpha>0$, where the tangent is parallel to $y=x$ has an ordinate $\frac{u^{2}}{4a}$, then what is the value of $\alpha$?

Question 2:

Prove that the segment of the tangent to the curve $y=c/x$, which is contained between the coordinate axes is bisected at the point of tangency.

Question 3:

Find all the tangents to the curve $y = \cos{(x+y)}$ for $-\pi \leq x \leq \pi$ that are parallel to the line $x+2y=0$.

Question 4:

Prove that the curves $y=f(x)$, where $f(x)>0$, and $y=f(x)\sin{x}$, where $f(x)$ is a differentiable function have common tangents at common points.

Question 5:

Find the condition that the lines $x \cos{\alpha} + y \sin{\alpha} = p$ may touch the curve $(\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1$.

Question 6:

Find the equation of a straight line which is tangent to one point and normal to the point on the curve $y=8t^{3}-1$, and $x=4t^{2}+3$.

Question 7:

Three normals are drawn from the point $(c,0)$ to the curve $y^{2}=x$. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

Question 8:

If $p_{1}$ and $p_{2}$ are lengths of the perpendiculars from origin on the tangent and normal to the curve $x^{2/3} + y^{2/3}=a^{2/3}$ respectively, prove that $4p_{1}^{2} + p_{2}^{2}=a^{2}$.

Question 9:

Show that the curve $x=1-3t^{2}$, and $y=t-3t^{3}$ is symmetrical about x-axis and has no real points for $x>1$. If the tangent at the point t is inclined at an angle $\psi$ to OX, prove that $3t= \tan {\psi} +\sec {\psi}$. If the tangent at $P(-2,2)$ meets the curve again at Q, prove that the tangents at P and Q are at right angles.

Question 10:

Find the condition that the curves $ax^{2}+by^{2}=1$ and $a^{'}x^{2} + b^{'}y^{2}=1$ intersect orthogonality and hence show that the curves $\frac{x^{2}}{(a^{2}+b_{1})} + \frac{y^{2}}{(b^{2}+b_{1})} = 1$ and $\frac{x^{2}}{a^{2}+b_{2}} + \frac{y^{2}}{(b^{2}+b_{2})} =1$ also intersect orthogonally.

More later,

Nalin Pithwa.