Calculus Made Easy · Silvanus P. Thompson

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Exercises I. ( p. 24 .)

(1) (d y)/(d x)=13 x^(12) .

(2) (d y)/(d x)=-3/2 x^(-5/2) .

(3) (d y)/(d x)=2 a x^((2 a-1)) .

(4) (d u)/(d t)=2.4 t^(1.4) .

(5) (d z)/(d u)=1/3 u^(-2/3) .

(6) (d y)/(d x)=-5/3 x^(-8/3) .

(7) (d u)/(d x)=-8/5 x^(-13/5) .

(8) (d y)/(d x)=2 a x^(a-1) .

(9) (d y)/(d x)=3/q x^((3-q)/q) .

(10) (d y)/(d x)=-m/n x^(-(m+n)/n) .

Exercises II. ( p. 31 .)

(1) (d y)/(d x)=3 a x^2 .

(2) (d y)/(d x)=13 × 3/2 x^(1/2) .

(3) (d y)/(d x)=6 x^(-1/2) .

(4) (d y)/(d x)=1/2 c^(1/2) x^(-1/2) .

(5) (d u)/(d z)=(a n)/c z^(n-1) .

(6) (d y)/(d t)=2.36 t .

(7) (d l_t)/(d t)=0.000012 × l_0 .

(8) (d C)/(d V)=a b V^(b-1), 0.98,3.00 and 7.47 candle power per volt respectively.

(9) (d n)/(d D)=-1/(L D^2) √((g T)/(π σ)), (d n)/(d L)=-1/(D L^2) √((g T)/(π σ)) , (d n)/(d σ)=-1/(2 D L) √((g T)/(π σ^3)), (d n)/(d T)=1/(2 D L) √(g/(π σ T)) .

(10) (Rate of change of P when t varies)/(Rate of change of P when D varies)=-D/t .

(11) 2 π, 2 π r, π l, 2/3 π r h, 8 π r, 4 π r^2 .

(12) (d D)/(d T)=(0.000012 l_t)/π .

Exercises III. ( p. 45 .)

(1) (a) 1+x+(x^2)/2+(x^3)/6+(x^4)/24+… .

(b) 2 a x+b .

(c) 2 x+2 a .

(d) 3 x^2+6 a x+3 a^2 .

(2) (d w)/(d t)=a-b t .

(3) (d y)/(d x)=2 x .

(4) 14110 x^4-65404 x^3-2244 x^2+8192 x+1379 .

(5) (d x)/(d y)=2 y+8 .

(6) 185.9022654 x^2+154.36334 .

(7) (-5)/((3 x+2)^2) .

(8) (6 x^4+6 x^3+9 x^2)/((1+x+2 x^2)^2) .

(9) (a d-b c)/((c x+d)^2) .

(10) (a n x^(-n-1)+b n x^(n-1)+2 n x^(-1))/((x^(-n)+b)^2) .

(11) b+2 c t .

(12) R_0(a+2 b t), R_0(a+b/(2 √(t))), -(R_0(a+2 b t))/((1+a t+b t^2)^2) or (R^2(a+2 b t))/R_0 .

(13) 1.4340(0.000014 t-0.001024),-0.00117,-0.00107,-0.00097 .

(14) (d E)/(d l)=b+k/i, (d E)/(d i)=-(c+k l)/(i^2) .

Exercises IV. ( p. 50 .)

(1) 17+24 x; 24 .

(2) (x^2+2 a x-a)/((x+a)^2); (2 a(a+1))/((x+a)^3) .

(3) 1+x+(x^2)/(1 × 2)+(x^3)/(1 × 2 × 3); 1+x+(x^2)/(1 × 2) .

(4) ( Exercises III. ):

(1) (a) (d^2 y)/(d x^2)=(d^3 y)/(d x^3)=1+x+1/2 x^2+1/6 x^3+… .

(b) 2 a, 0 .

(c) 2,0.

(d) 6 x+6 a, 6 .

(2) -b, 0 .

(3) 2,0.

(4) 56440 x^3-196212 x^2-4488 x+8192 . 169320 x^2-392424 x-4488.

(5) 2,0.

(6) 371.80453 x, 371.80453 .

(7) 30/((3 x+2)^3), -270/((3 x+2)^4) .

( Examples , p. 40):

(1) (6 a)/(b^2) x, (6 a)/(b^2) .

(2) (3 a √(b))/(2 √(x))-dfrac6 b sqrt[3]ax^3, dfrac18 b sqrt[3]ax^4-(3 a √(b))/(4 √(x^3)) .

(3) dfrac2sqrt[3]θ^8-dfrac1.056sqrt[5]θ^(11), dfrac2.3232sqrt[5]θ^(16)-dfrac163 sqrt[3]θ^(11) .

(4) 810 t^4-648 t^3+479.52 t^2-139.968 t+26.64 . 3240 t^3-1944 t^2+959.04 t-139.968.

(5) 12 x+2,12 .

(6) 6 x^2-9 x, 12 x-9 .

(7) 3/4(1/√(θ)+1/(√(θ^5)))+1/4(15/(√(θ^7))-1/(√(θ^3))) . 3/8(1/(√(θ^5))-1/(√(θ^3)))-15/8(7/(√(θ^9))+1/(√(θ^7))).

Exercises V. ( p. 63 .)

(2) 64; 147.2; and 0.32 feet per second.

(3) x=a-g t; xddot=-g .

(4) 45.1 feet per second.

(5) 12.4 feet per second per second. Yes.

(6) Angular velocity =11.2 radians per second; angular acceleration =9.6 radians per second per second.

(7) v=20.4 t^2-10.8. a=40.8 t. 172.8 in./sec., 122.4 in./sec.^2 .

(8) v=dfrac130 sqrt[3](t-125)^2, a=-dfrac145 sqrt[3](t-125)^5 .

(9) v=0.8-(8 t)/((4+t^2)^2), a=(24 t^2-32)/((4+t^2)^3), 0.7926 and 0.00211.

(10) n=2, n=11 .

Exercises VI. ( p. 72 .)

(1) x/(√(x^2+1)) .

(2) x/(√(x^2+a^2)) .

(3) -1/(2 √((a+x)^3)) .

(4) (a x)/(√((a-x^2)^3)) .

(5) (2 a^2-x^2)/(x^3 √(x^2-a^2)) .

(6) dfrac3/2 x^2[8/9 x(x^3+a)-(x^4+a)] (x^4+a)^(2/3)(x^3+a)^(3/2) .

(7) (2 a(x-a))/((x+a)^3) .

(8) 5/2 y^3 .

(9) 1/((1-θ) √(1-θ^2)) .

Exercises VII. ( p. 74 .)

(1) (d w)/(d x)=(3 x^2(3+3 x^3))/(27(1/2 x^3+1/4 x^6)^3) .

(2) (d v)/(d x)=-(12 x)/(√(1+√(2)+3 x^2)(√(3)+4 √(1+√(2)+3 x^2))^2) .

(3) (d u)/(d x)=-(x^2(√(3)+x^3))/(√([1+(1+(x^3)/√(3))^2]^3)) .

Exercises VIII. ( p. 88 .)

(2) 1.44.

(4) (d y)/(d x)=3 x^2+3 ; and the numerical values are: 3,33/4, 6 , and 15.

(5) ± √(2) .

(6) (d y)/(d x)=-4/9 x/y . Slope is zero where x=0 ; and is mp 1/(3 √(2)) where x=1 .

(7) m=4, n=-3 .

(8) Intersections at x=1, x=-3 . Angles 153^(°) 26^(′), 2^(°) 28^(′) .

(9) Intersection at x=3.57, y=3.50 . Angle 16^(°) 16^(′) .

(10) x=1/3, y=2 1/3, b=-5/3 .

Exercises IX. ( p. 107 .)

(1) Min.: x=0, y=0 ; max.: x=-2, y=-4 .

(2) x=a .

(4) 25 √(3) square inches.

(5) (d y)/(d x)=-10/(x^2)+10/((8-x)^2); x=4; y=5 .

(6) Max. for x=-1 ; min. for x=1 .

(7) Join the middle points of the four sides.

(8) r=2/3 R, r=R/2 , no max.

(9) r=R √(2/3), r=R/√(2), r=0.8506 R .

(10) At the rate of 8/r square feet per second.

(11) r=(R √(8))/3 .

(12) n=√((N R)/r) .

Exercises X. ( p. 115 .)

(1) Max.: x=-2.19, y=24.19 ; min.: x=1.52, y=-1.38 .

(2) (d y)/(d x)=b/a-2 c x; (d^2 y)/(d x^2)=-2 c; x=b/(2 a c) (a maximum).

(3) (a) One maximum and two minima.

(b) One maximum. ( x=0 ; other points unreal.)

(4) Min.: x=1.71, y=6.14 .

(5) Max: x=-.5, y=4 .

(6) Max.: x=1.414, y=1.7675 .

Min.: x=-1.414, y=1.7675 .

(7) Max.: x=-3.565, y=2.12 .

Min.: x=+3.565, y=7.88 .

(8) 0.4 N, 0.6 N .

(9) x=√(a/c) .

(10) Speed 8.66 nautical miles per hour. Time taken 115.47 hours. Minimum cost £112.12s.

(11) Max. and min. for x=7.5, y= ± 5.414 . (See example no. 10, p. 71.)

(12) Min.: x=1/2, y=0.25 ; max.: x=-1/3, y=1.408 .

Exercises XI. ( p. 127 .)

(1) 2/(x-3)+1/(x+4) .

(2) 1/(x-1)+2/(x-2) .

(3) 2/(x-3)+1/(x+4) .

(4) 5/(x-4)-4/(x-3) .

(5) 19/(13(2 x+3))-22/(13(3 x-2)) .

(6) 2/(x-2)+4/(x-3)-5/(x-4) .

(7) 1/(6(x-1))+11/(15(x+2))+1/(10(x-3)) .

(8) 7/(9(3 x+1))+71/(63(3 x-2))-5/(7(2 x+1)) .

(9) 1/(3(x-1))+(2 x+1)/(3(x^2+x+1)) .

(10) x+2/(3(x+1))+(1-2 x)/(3(x^2-x+1)) .

(11) 3/((x+1))+(2 x+1)/(x^2+x+1) .

(12) 1/(x-1)-1/(x-2)+2/((x-2)^2) .

(13) 1/(4(x-1))-1/(4(x+1))+1/(2(x+1)^2) .

(14) 4/(9(x-1))-4/(9(x+2))-1/(3(x+2)^2) .

(15) 1/(x+2)-(x-1)/(x^2+x+1)-1/((x^2+x+1)^2) .

(16) 5/(x+4)-32/((x+4)^2)+36/((x+4)^3) .

(17) 7/(9(3 x-2)^2)+55/(9(3 x-2)^3)+73/(9(3 x-2)^4) .

(18) 1/(6(x-2))+1/(3(x-2)^2)-x/(6(x^2+2 x+4)) .

Exercises XII. ( p. 150 .)

(1) a b(ε^(a x)+ε^(-a x)) .

(2) 2 a t+2/t .

(3) log _(ε) n .

(5) n p v^(n-1) .

(6) n/x .

(7) (3 ε^(-x/(x-1)))/((x-1)^2) .

(8) 6 x ε^(-5 x)-5(3 x^2+1) ε^(-5 x) .

(9) (a x^(a-1))/(x^a+a) .

(10) ((6 x)/(3 x^2-1)+1/(2(√(x)+x)))(3 x^2-1)(√(x)+1) .

(11) (1- log _(ε)(x+3))/((x+3)^2) .

(12) a^x(a x^(a-1)+x^a log _(ε) a) .

(14) Min.: y=0.7 for x=0.694 .

(15) (1+x)/x .

(16) 3/x(log _(ε) a x)^2 .

Exercises XIII. ( p. 160 .)

(1) Let t/T=x(∴ t=8 x) , and use the Table on page 157 .

(2) T=34.627; 159.46 minutes.

(3) Take 2 t=x ; and use the Table on page 157 .

(5) (a) x^x(1+ log _(ε) x) ;

(b) 2 x(ε^x)^x ;

(c) ε^x^x × x^x(1+ log _(ε) x) .

(6) 0.14 second.

(7) (a) 1.642; (b) 15.58.

(8) μ=0.00037,31^m 1/4 .

(9) i is 63.4 \% of i_0, 220 kilometres.

(10) 0.133,0.145,0.155 , mean 0.144;-10.2 \%,-0.9 \%,+77.2 \% .

(11) Min. for x=1/ε .

(12) Max. for x=ε .

(13) Min. for x= log _(ε) a .

Exercises XIV. ( p. 170 .)

(1) (i) (d y)/(d θ)=A cos (θ-π/2) ;

(ii) (d y)/(d θ)=2 sin θ cos θ= sin 2 θ and (d y)/(d θ)=2 cos 2 θ ;

(iii) (d y)/(d θ)=3 sin ^2 θ cos θ and (d y)/(d θ)=3 cos 3 θ .

(2) θ=45^(°) or π/4 radians.

(3) (d y)/(d t)=-n sin 2 π n t .

(4) a^x log _(ε) a cos a^x .

(5) (cos x)/(sin x)= cotan x .

(6) 18.2 cos (x+26^(°)) .

(7) The slope is (d y)/(d θ)=100 cos (θ-15^(°)) , which is a maximum when (θ-15^(°))=0 , or θ=15^(°) ; the value of the slope being then =100 . When θ=75^(°) the slope is 100 cos (75^(°)-15^(°))=100 cos 60^(°)= 100 × 1/2=50 .

(8) cos θ sin 2 θ+2 cos 2 θ sin θ =2 sin θ(cos ^2 θ+ cos 2 θ) ; =2 sin θ(3 cos ^2 θ-1).

(9) a m n θ^(n-1) tan ^(m-1)(θ^n) sec ^2 θ^n .

(10) ε^x(sin ^2 x+ sin 2 x); ε^x(sin ^2 x+2 sin 2 x+2 cos 2 x).

(11) (i) (d y)/(d x)=(a b)/((x+b)^2) ;

(ii) a/b ε^(-x/b) ;

(iii) 1/90 ° × (a b)/((b^2+x^2)) .

(12) (i) (d y)/(d x)= sec x tan x ;

(ii) (d y)/(d x)=-1/(√(1-x^2)) ;

(iii) (d y)/(d x)=1/(1+x^2) ;

(iv) (d y)/(d x)=1/(x √(x^2-1)) ;

(v) (d y)/(d x)=(√(3 sec x)(3 sec ^2 x-1))/2 .

(13) (d y)/(d θ)=4.6(2 θ+3)^(1.3) cos (2 θ+3)^(2.3) .

(14) (d y)/(d θ)=3 θ^2+3 cos (θ+3)- log _(ε) 3(cos θ × 3^(sin θ)+3 θ) .

(15) θ= cot θ; θ= ± 0.86 ; is max. for +θ , min. for -θ .

Exercises XV. ( p. 177 .)

(1) x^3-6 x^2 y-2 y^2; 1/3-2 x^3-4 x y .

(2) 2 x y z+y^2 z+z^2 y+2 x y^2 z^2 ; 2 x y z+x^2 z+x z^2+2 x^2 y z^2; ; 2 x y z+x^2 y+x y^2+2 x^2 y^2 z.

(3) 1/r\(x-a)+(y-b)+(z-c)\=((x+y+z)-(a+b+c))/r; 3/r .

(4) d y=v u^(v-1) d u+u^v log _(ε) u d v .

(5) d y =3 sin v u^2 d u+u^3 cos v d v, ; d y =u sin x^(u-1) cos x d x+(sin x)^u log _(ε) sin x d u, ; d y =1/v 1/u d u- log _(ε) u 1/(v^2) d v.

(7) Minimum for x=y=-1/2 .

(8) (a) Length 2 feet, width = depth =1 foot, vol. =2 cubic feet. (b) Radius =2/π feet =7.46 in., length =2 feet, vol. =2.54.

(9) All three parts equal; the product is maximum.

(10) Minimum for x=y=1 .

(11) Min.: x=1/2 and y=2 .

(12) Angle at apex =90^(°) ; equal sides = length =sqrt[3]2 V .

Exercises XVI. ( p. 187 .)

(1) 1 1/3 .

(2) 0.6344.

(3) 0.2624.

(4) (a) y=1/8 x^2+C ;

(b) y= sin x+C .

(5) y=x^2+3 x+C .

Exercises XVII. ( p. 202 .)

(1) (4 √(a) x^(3/2))/3+C .

(2) -1/(x^3)+C .

(3) (x^4)/(4 a)+C .

(4) 1/3 x^3+a x+C .

(5) -2 x^(-5/2)+C .

(6) x^4+x^3+x^2+x+C .

(7) (a x^2)/4+(b x^3)/9+(c x^4)/16+C .

(8) (x^2+a)/(x+a)=x-a+(a^2+a)/(x+a) by division. Therefore the answer is (x^2)/2-a x+(a^2+a) log _(ε)(x+a)+C . (See pages 196 and 198 .)

(9) (x^4)/4+3 x^3+27/2 x^2+27 x+C .

(10) (x^3)/3+(2-a)/2 x^2-2 a x+C .

(11) a^2(2 x^(3/2)+9/4 x^(4/3))+C .

(12) -1/3 cos θ-1/6 θ+C .

(13) θ/2+(sin 2 a θ)/(4 a)+C .

(14) θ/2-(sin 2 θ)/4+C .

(15) θ/2-(sin 2 a θ)/(4 a)+C .

(16) 1/3 ε^(3 x) .

(17) log (1+x)+C .

(18) - log _(ε)(1-x)+C .

Exercises XVIII. ( p. 221 .)

(1) Area =60 ; mean ordinate =10 .

(2) Area =2/3 of a × 2 a √(a) .

(3) Area =2 ; mean ordinate =2/π=0.637 .

(4) Area =1.57 mean ordinate =0.5 .

(5) 0.572,0.0476 .

(6) Volume =π r^2 h/3 .

(7) 1.25.

(8) 79.4.

(9) Volume =4.9348 ; area of surface =12.57 (from 0 to π ).

(10) a log _(ε) a, a/(a-1) log _(ε) a .

(12) Arithmetical mean =9.5 ; quadratic mean =10.85 .

(13) Quadratic mean =1/√(2) √(A_1^2+A_3^2) ; arithmetical mean =0 .

The first involves a somewhat difficult integral, and may be stated thus: By definition the quadratic mean will be √(1/(2 π) ∫_0^(2 π)(A_1 sin x+A_3 sin 3 x)^2 d x).

Now the integration indicated by ∫(A_1^2 sin ^2 x+2 A_1 A_3 sin x sin 3 x+A_3^2 sin ^2 3 x) d x is more readily obtained if for sin ^2 x we write (1- cos 2 x)/2. For 2 sin x sin 3 x we write cos 2 x- cos 4 x ; and, for sin ^2 3 x , (1- cos 6 x)/2.

Making these substitutions, and integrating, we get (see p. 198 ) (A_1^2)/2(x-(sin 2 x)/2)+A_1 A_3((sin 2 x)/2-(sin 4 x)/4)+(A_3^2)/2(x-(sin 6 x)/6).

At the lower limit the substitution of 0 for x causes all this to vanish, whilst at the upper limit the substitution of 2 π for x gives A_1^2 π+A_3^2 π . 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) (x √(a^2-x^2))/2+(a^2)/2 sin ^(-1) x/a+C .

(2) (x^2)/2(log _(ε) x-1/2)+C .

(3) (x^(a+1))/(a+1)(log _(ε) x-1/(a+1))+C .

(4) sin ε^x+C .

(5) sin (log _(ε) x)+C .

(6) ε^x(x^2-2 x+2)+C .

(7) 1/(a+1)(log _(ε) x)^(a+1)+C .

(8) log _(ε)(log _(ε) x)+C .

(9) 2 log _(ε)(x-1)+3 log _(ε)(x+2)+C .

(10) 1/2 log _(ε)(x-1)+1/5 log _(ε)(x-2)+3/10 log _(ε)(x+3)+C .

(11) b/(2 a) log _(ε) (x-a)/(x+a)+C .

(12) log _(ε) (x^2-1)/(x^2+1)+C .

(13) 1/4 log _(ε) (1+x)/(1-x)+1/2 arctan x+C .

(14) 1/√(a) log _(ε) (√(a)-√(a-b x^2))/(x √(a)) . (Let 1/x=v ; then, in the result, let √(v^2-b/a)=v-u . )

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 .

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