**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.