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DIFFERENTIAL CALCULUS.

CXXV.

1. What is a function of an independent variable? Explain what is meant by the differential coefficient of such a function.

2. Exhibit, without proving it, the ordinary form of Taylor's theorem, and deduce from it the theorem known as Maclaurin’s.

3. Expand sin x in terms of x to three terms by Maclaurin's theorem; and express the general term of the series for sin x.

4. If a tangent be drawn at any point (x, y) of a plane curve, examine the trigonometrical meaning of

reference to that tangent.

dy

dx

with

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9. Find the subtangent of a spiral curve from its polar equation, and show how to determine whether the curve has an asymptote or not.

Ex. Find the subtangent and asymptote of the curve

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11. Expand tan-1x in a series of ascending powers of x by Maclaurin's theorem.

12. Find what the area of the base of the roomiest tent of a conical form will be, which can be covered by 200 feet of canvas.

13. Explain how the form of a plane curve is determined by means of an equation, and illustrate the explanation by reference to some particular curve.

14. Find the differential coefficient in respect to x of xm. 15. Find the equation to the tangent drawn to any point of the locus x2 + y2=c2, and prove that it is always perpendicular to the radius vector.

16. If be the angle which the normal to a curve makes with the radius vector r, from the pole at the point (r, 0), prove that sin : &= cos :

dr
ds'

rdė

ds

17. Explain and illustrate the meaning of the limit of a variable ratio, and show how such a limit is applied in the fundamental principles of the differential calculus.

18. Investigate the differential coefficient of sin-1ax with respect to x, and differentiate—

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(3) (tan x) cot1x; (4) cos {cos (cos x)}.

19. If y=(a+x)3 (a-x), find the values of x which make y a maximum or a minimum.

20. Determine the position and magnitude of the radius of curvature of the curve √x+ √√a, at the point for

which x and y are each equal to

21. Expand as far as a

α

(1) log (x+√x2 + a2) · (?) (e2 +e ̄x)".

22. Prove the following differential formulæ which occur in the theory of plane curves :

dy

dx

(1) Subtangent=y; (2) subnormal=y;

dx

dy

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24. Determine the maxima and minima off (x), when f(x)=(x-2) (x-4)2.

25. Determine the ratio between the height and radius of the base of a cylindrical quart cup so that its surface may be a minimum.

26. Show that if y=f (x) be the equation to a curve, the equation to the tangent is y-y=ƒ' (x') (x—x').

27. Find the maximum or minimum value of

3x+4x3-6x-12x+13.

28. Find the nth differential coefficients of (1) x2 log x, and (2) x3ex.

29. If u=(x+1) (x2+1), find the value of

du

dx

30. Find the values of x which will make

sin (x-a) cos x a maximum or minimum.

31. Supposing r=r cos 0, y=r sin 0, find the value of xdy-ydx in terms of r and 0.

32. Define the differential coefficient of a function, and investigate the differential coefficients with respect to x of ax and of tan ax.

33. Expand ex cos x, by Maclaurin's theorem, to 3 terms, writing down the general term.

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develope sin x and cos x in ascending powers of x, by the

method of indeterminate coefficients.

35. Expand to 4 terms in powers of x, by Maclaurin's theorem, ecos

36. Obtain the equation of the tangent to the curve (x, y)=0, and show that it is one dimension lower, at least, than the equation to the curve.

37. Prove that

d (tan x)_sec2 x.

dx

38. Find the differentials of the following expressions :(1) cos √x2+a2; (2) sin √x2+a2; (3) log √x2+a2. 39. A given sum £a is to be laid out in the purchase of a rectangular piece of ground and in building a wall round it. The wall on one side will cost £m per lineal foot, and on the other three sides £n per foot; and the land will cost £p per square foot. How may the greatest piece of land be enclosed subject to these conditions?

40. Find the differential coefficient of √a2-x2 with respect to x by first principles.

41. Find the values of x which make x (a−x)2 (2a—x)3 a maximum or a minimum.

42. Find the value of the ralius of curvature at any point of a curve; for example, a parabola at its vertex.

43. Expand e log, (1+x) by Maclaurin's theorem. 44. Find the real value of

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с

; (2) y=ce"; (3) y=
(3) y=2(e2+e3).

48. Prove that the greatest rectangle which can be inscribed in an ellipse is half of that contained by the axes. 49. If r be the radius vector at any point of a curve, and the perpendicular from the pole upon the tangent, prove the radius of curvature equal to

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Ρ

rdr

dp

In the ellipse, whose semi-axes are a, b, given p2=

b2r

2a-r'

find the radius of curvature drawn at the extremity of the major axis.

50. Two sides of a triangle being of given lengths, show that its area will be a maximum when they contain a right angle.

51. Hence, prove that if all the sides except one of a plane rectilineal figure be given, its area will be a maximum when the remaining side is the diameter of a semicircle circumscribing the figure.

52. And consequently if all the sides of a rectilineal polygon be given in length, its area will be a maximum when it admits of being inscribed in a circle.

53. Assuming the differential equation to the tangent to a plane curve referred to rectangular coordinates, show how the asymptotes to the curve may be drawn, if it has any.

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54. Investigate the conditions under which a function of a single variable becomes a maximum or a minimum.

55. A rectangular plot of ground of given area is to be enclosed from the waste by a wall and divided into three equal areas, by partition walls parallel to one of its sides. What must be the dimensions of the rectangle that the length of walling may be a minimum ?

56. Expand, by Maclaurin's theorem, tan1 x in terms of x to 3 terms.

57. Expand (1+2x+322) to 5 terms by Maclaurin's theorem.

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