Math moments: uses of mathematics in today’s world

Derivatives: IITJEE Mains Maths: Mathematics Hothouse video lecture

Limits Part 2: IITJEE Mains maths: Mathematics Hothouse

Limits part 1: video lecture: IITJEE Mains maths: Mathematics Hothouse

 

IIT JEE Foundation video lecture: Plane Geometry: From Mathematics Hothouse

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.