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

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

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

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

and that

find the value of

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

Expand 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.

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

at any point, prove that 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:—

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.

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9. Find the following integrals:—

10. Find the length of the arc of the curve between the axes of coordinates.

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