Time is Life.

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

Mathematics demystified

November 27, 2018 – 10:40 pm

Time is Life.

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

November 27, 2018 – 10:36 pm

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. *

November 27, 2018 – 10:07 pm

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.

November 13, 2018 – 3:16 pm

**Question I:**

Show that the equation of the tangent to the curve and can be represented in the form:

**Question 2:**

Show that the derivative of the function , when and , when vanishes on an infinite set of points of the interval $latex (0,1), Hint: Use Rolle’s theorem.

**Question 3:**

Prove that for . Use Lagrange’s theorem.

**Question 4:**

Find the largest term in the sequence . Hint: Consider the function in the interval .

**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 of degree 6, which satisfies and has a local maximum at and local minimum at and 2.

**Question 7:**

For the circle , find the value of r for which the area enclosed by the tangents drawn from the point 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 , with for all x. Prove that .

**Question 9:**

Show that , if .

**Question 10:**

Let . Show that the equations has a unique root in the interval and identify it.

November 13, 2018 – 6:26 am

**Question 1.**

If the point on , where , where the tangent is parallel to has an ordinate , then what is the value of ?

**Question 2:**

Prove that the segment of the tangent to the curve , which is contained between the coordinate axes is bisected at the point of tangency.

**Question 3:**

Find all the tangents to the curve for that are parallel to the line .

**Question 4:**

Prove that the curves , where , and , where is a differentiable function have common tangents at common points.

**Question 5:**

Find the condition that the lines may touch the curve .

**Question 6:**

Find the equation of a straight line which is tangent to one point and normal to the point on the curve , and .

**Question 7:**

Three normals are drawn from the point to the curve . 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 and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .

**Question 9:**

Show that the curve , and is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at 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 and intersect orthogonality and hence show that the curves and also intersect orthogonally.

More later,

Nalin Pithwa.