and HONOURS. Calculus, I. 1. Assuming the rule for differentiating pq where p are functions of x, establish the rule for differentia 2. (a) Establish the rule for differentiating log p. du (b) Given u=log (x +Va2+x2) to find dx 3. (a) Obtain a differential expression for the tangent to the curve y=f(x) at the point (x, y). (b) Show that in the curve x3+y3; a3, the part of the tangent intercepted between the axes is constant. 4. In the polar curve r=f(0)) find the angle which the tangent at the point (r, 0) makes with (a) the prime vector; (b) the radius vector. 5. (a) If y=f(x) be a curve give the geometrical meaning of 12 ց dx 2 (b) Give the conditions that f(x) may have a maximum or a minimum value, drawing your conclusions from the theory of a curve. (c) Find the least ellipse described about a given rectangle, and show that its area is to that of the rectangle as π: 2. 6. Write the expression for the radius of the osculating circle of any curve; and show that in any conic this radius is the cube of the normal divided by the square of the semi-latus rectum. (b) Given ƒ(x) to develope ƒ(117) 8. Prove that the general term in the expansion of (sin-1x)2 is 10. Prove that the whole area of the curve r2a2 cos 20 is a2. |