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.