ANSWERS.
Reading edition
Exercises I. (p. 24.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) .
(9) .
(10) .
Exercises II. (p. 31.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) and 7.47 candle power per volt respectively.
(9) , .
(10) .
(11) .
(12) .
Exercises III. (p. 45.)
(1) (a) .
(b) .
(c) .
(d) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) .
(9) .
(10) .
(11) .
(12) or .
(13) .
(14) .
Exercises IV. (p. 50.)
(1) .
(2) .
(3) .
(4) (Exercises III.):
(1) (a) .
(b) .
(c) 2,0.
(d) .
(2) .
(3) 2,0.
(4) .
(5) 2,0.
(6) .
(7) .
(Examples, p. 40):
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
Exercises V. (p. 63.)
(2) 64; 147.2; and 0.32 feet per second.
(3) .
(4) 45.1 feet per second.
(5) 12.4 feet per second per second. Yes.
(6) Angular velocity radians per second; angular acceleration radians per second per second.
(7) .
(8) .
(9) and 0.00211.
(10) .
Exercises VI. (p. 72.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) .
(9) .
Exercises VII. (p. 74.)
(1) .
(2) .
(3) .
Exercises VIII. (p. 88.)
(2) 1.44.
(4) ; and the numerical values are: , and 15.
(5) .
(6) . Slope is zero where ; and is where .
(7) .
(8) Intersections at . Angles .
(9) Intersection at . Angle .
(10) .
Exercises IX. (p. 107.)
(1) Min.: ; max.: .
(2) .
(4) square inches.
(5) .
(6) Max. for ; min. for .
(7) Join the middle points of the four sides.
(8) , no max.
(9) .
(10) At the rate of square feet per second.
(11) .
(12) .
Exercises X. (p. 115.)
(1) Max.: ; min.: .
(2) (a maximum).
(3) (a) One maximum and two minima.
(b) One maximum. ( ; other points unreal.)
(4) Min.: .
(5) Max: .
(6) Max.: .
Min.: .
(7) Max.: .
Min.: .
(8) .
(9) .
(10) Speed 8.66 nautical miles per hour. Time taken 115.47 hours. Minimum cost £112.12s.
(11) Max. and min. for . (See example no. 10, p. 71.)
(12) Min.: ; max.: .
Exercises XI. (p. 127.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) .
(9) .
(10) .
(11) .
(12) .
(13) .
(14) .
(15) .
(16) .
(17) .
(18) .
Exercises XII. (p. 150.)
(1) .
(2) .
(3) .
(5) .
(6) .
(7) .
(8) .
(9) .
(10) .
(11) .
(12) .
(14) Min.: for .
(15) .
(16) .
Exercises XIII. (p. 160.)
(1) Let , and use the Table on page 157.
(2) minutes.
(3) Take ; and use the Table on page 157.
(5) (a) ;
(b) ;
(c) .
(6) 0.14 second.
(7) (a) 1.642; (b) 15.58.
(8) .
(9) is of kilometres.
(10) , mean .
(11) Min. for .
(12) Max. for .
(13) Min. for .
Exercises XIV. (p. 170.)
(1) (i) ;
(ii) and ;
(iii) and .
(2) or radians.
(3) .
(4) .
(5) .
(6) .
(7) The slope is , which is a maximum when , or ; the value of the slope being then . When the slope is .
(8)
(9) .
(10)
(11) (i) ;
(ii) ;
(iii) .
(12) (i) ;
(ii) ;
(iii) ;
(iv) ;
(v) .
(13) .
(14) .
(15) ; is max. for , min. for .
Exercises XV. (p. 177.)
(1) .
(2) ;
(3) .
(4) .
(5)
(7) Minimum for .
(8) (a) Length 2 feet, width depth foot, vol. cubic feet.
(9) All three parts equal; the product is maximum.
(10) Minimum for .
(11) Min.: and .
(12) Angle at apex ; equal sides length .
Exercises XVI. (p. 187.)
(1) .
(2) 0.6344.
(3) 0.2624.
(4) (a) ;
(b) .
(5) .
Exercises XVII. (p. 202.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) by division. Therefore the answer is . (See pages 196 and 198.)
(9) .
(10) .
(11) .
(12) .
(13) .
(14) .
(15) .
(16) .
(17) .
(18) .
Exercises XVIII. (p. 221.)
(1) Area ; mean ordinate .
(2) Area of .
(3) Area ; mean ordinate .
(4) Area mean ordinate .
(5) .
(6) Volume .
(7) 1.25.
(8) 79.4.
(9) Volume ; area of surface (from 0 to ).
(10) .
(12) Arithmetical mean ; quadratic mean .
(13) Quadratic mean ; arithmetical mean .
The first involves a somewhat difficult integral, and may be stated thus: By definition the quadratic mean will be
Now the integration indicated by is more readily obtained if for we write For we write ; and, for ,
Making these substitutions, and integrating, we get (see p. 198)
At the lower limit the substitution of 0 for causes all this to vanish, whilst at the upper limit the substitution of for gives . And hence the answer follows.
(14) Area is 62.6 square units. Mean ordinate is 10.42.
(16) 436.3. (This solid is pear shaped.)
Exercises XIX. (p. 231.)
(1) .
(2) .
(3) .
(4) .
(5) .
(6) .
(7) .
(8) .
(9) .
(10) .
(11) .
(12) .
(13) .
(14) . (Let ; then, in the result, let .)
You had better differentiate now the answer and work back to the given expression as a check.
Every earnest student is exhorted to manufacture more examples for himself at every stage, so as to test his powers. When integrating he can always test his answer by differentiating it, to see whether he gets back the expression from which he started.
There are lots of books which give examples for practice. It will suffice here to name two: R. G. Blaine's The Calculus and its Applications, and F. M. Saxelby's A Course in Practical Mathematics.