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