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The Pamphlet Collection of Sir Robert Stout: Volume 50

University of New Zealand — Paper c. Differential and Integral Calculus. — Examiner: Prof. C. Niven, F.R.S

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University of New Zealand

Paper c. Differential and Integral Calculus.

Examiner: Prof. C. Niven, F.R.S.

1. Investigate the differential coefficient of cos x, and differentiate

formula/equation

2. State and prove Leibnitz's Theorem. Assuming that

formula/equation

and that

formula/equation

find the value of

formula/equation

page 2

3. State and prove Taylor's Theorem, including the expression for the remainder after n terms.

Expand formula/equation to 4 terms.

4. Find the ellipse of greatest area which can be inscribed in a given isosceles triangle, so as to have one axis coinciding with the line which bisects the vertical angle of the triangle.

formula/equation

6. If Φ be the angle between a curve and the radius vector

at any point, prove that formula/equation Find also the polar equation of the tangent.

In the curve r2 = a2 sin 2θ, the tangent at P meets the initial line Sx in T, and PN is perpendicular to Sx. Prove that

SP · ST = 2PT · SN.

7. Find the asymptotes of the following curves:—

formula/equation

Trace the curves.

8. Prove that the locus of the centres of curvature of a curve is the envelope of its normals. Why is this locus called the evolute?

Find the evolute of the parabola.

page 3

9. Find the following integrals:—

formula/equation

10. Find the length of the arc of the curve formula/equation between the axes of coordinates.

11. Solve the differential equations formula/equation and find the curve in which the subnormal is equal to the sum of the ordinate and abscissa.