Calculus Made Easy/Answers to Exercises

​

(1) ${\displaystyle {\dfrac {dy}{dx}}=13x^{12}}$.

(2) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {3}{2}}x^{-{\frac {5}{2}}}}$.

(3) ${\displaystyle {\dfrac {dy}{dx}}=2ax^{(2a-1)}}$.

(4) ${\displaystyle {\dfrac {du}{dt}}=2.4t^{1.4}}$.

(5) ${\displaystyle {\dfrac {dz}{du}}={\dfrac {1}{3}}u^{-{\frac {2}{3}}}}$.

(6) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {5}{3}}x^{-{\frac {8}{3}}}}$.

(7) ${\displaystyle {\dfrac {du}{dx}}=-{\dfrac {8}{5}}x^{-{\frac {13}{5}}}}$.

(8) ${\displaystyle {\dfrac {dy}{dx}}=2ax^{a-1}}$.

(9) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {3}{q}}x^{\frac {3-q}{q}}}$.

(10) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {m}{n}}x^{-{\frac {m+n}{n}}}}$.

---

(1) ${\displaystyle {\dfrac {dy}{dx}}=3ax^{2}}$.

(2) ${\displaystyle {\dfrac {dy}{dx}}=13\times {\frac {3}{2}}x^{\frac {1}{2}}}$.

(3) ${\displaystyle {\dfrac {dy}{dx}}=6x^{-{\frac {1}{2}}}}$.

(4) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {1}{2}}c^{\frac {1}{2}}x^{-{\frac {1}{2}}}}$.

(5) ${\displaystyle {\dfrac {du}{dz}}={\dfrac {an}{c}}z^{n-1}}$.

(6) ${\displaystyle {\dfrac {dy}{dt}}=2.36t}$.

(7) ${\displaystyle {\dfrac {dl_{t}}{dt}}=0.000012\times l_{0}}$.

(8) ${\displaystyle {\dfrac {dC}{dV}}=abV^{b-1}}$, ${\displaystyle 0.98}$, ${\displaystyle 3.00}$ and ${\displaystyle 7.47}$ candle power per volt respectively.

(9) ${\displaystyle {\dfrac {dn}{dD}}=-{\dfrac {1}{LD^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}},{\dfrac {dn}{dL}}=-{\dfrac {1}{DL^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}}}$.
${\displaystyle \;\quad {\dfrac {dn}{d\sigma }}=-{\dfrac {1}{2DL}}{\sqrt {\dfrac {gT}{\pi \sigma ^{3}}}},{\dfrac {dn}{dT}}={\dfrac {1}{2DL}}{\sqrt {\dfrac {g}{\pi \sigma T}}}}$.

​(10) ${\displaystyle {\dfrac {\text{Rate of change of P when t varies}}{\text{Rate of change of P when D varies}}}=-{\dfrac {D}{t}}}$.

(11) ${\displaystyle 2\pi }$, ${\displaystyle 2\pi r}$, ${\displaystyle \pi l}$, ${\displaystyle {\tfrac {2}{3}}\pi rh}$, ${\displaystyle 8\pi r}$, ${\displaystyle 4\pi r^{2}}$.

(12) ${\displaystyle {\dfrac {dD}{dT}}={\dfrac {0.000012l_{t}}{\pi }}}$.

---

(1) (a) ${\displaystyle 1+x+{\dfrac {x^{2}}{2}}+{\dfrac {x^{3}}{6}}+{\dfrac {x^{4}}{24}}+\ldots }$.

(b) ${\displaystyle 2ax+b}$.

(c) ${\displaystyle 2x+2a}$.

(d) ${\displaystyle 3x^{2}+6ax+3a^{2}}$.

(2) ${\displaystyle {\dfrac {dw}{dt}}=a-bt}$.

(3) ${\displaystyle {\dfrac {dy}{dx}}=2x}$.

(4) ${\displaystyle 14110x^{4}-65404x^{3}-2244x^{2}+8192x+1379}$.

(5) ${\displaystyle {\dfrac {dx}{dy}}=2y+8}$.

(6) ${\displaystyle 185.9022654x^{2}+154.36334}$.

(7) ${\displaystyle {\dfrac {-5}{(3x+2)^{2}}}}$.

(8) ${\displaystyle {\dfrac {6x^{4}+6x^{3}+9x^{2}}{(1+x+2x^{2})^{2}}}}$.

(9) ${\displaystyle {\dfrac {ad-bc}{(cx+d)^{2}}}}$.

(10) ${\displaystyle {\dfrac {anx^{-n-1}+bnx^{n-1}+2nx^{-1}}{(x^{-n}+b)^{2}}}}$.

(11) ${\displaystyle b+2ct}$.

(12) ${\displaystyle R_{0}(a+2bt)}$, ${\displaystyle R_{0}\left(a+{\dfrac {b}{2{\sqrt {t}}}}\right)}$, ${\displaystyle -{\dfrac {R_{0}(a+2bt)}{(1+at+bt^{2})^{2}}}}$ or ${\displaystyle {\dfrac {R^{2}(a+2bt)}{R_{0}}}}$.

(13) ${\displaystyle 1.4340(0.000014t-0.001024)}$, ${\displaystyle -0.00117}$, ${\displaystyle -0.00107}$, ${\displaystyle -0.00097}$.

(14) ${\displaystyle {\dfrac {dE}{dl}}=b+{\dfrac {k}{i}}}$, ${\displaystyle {\dfrac {dE}{di}}=-{\dfrac {c+kl}{i^{2}}}}$.

---

(1) ${\displaystyle 17+24x}$; ${\displaystyle 24}$.

(2) ${\displaystyle {\dfrac {x^{2}+2ax-a}{(x+a)^{2}}}}$; ${\displaystyle {\dfrac {2a(a+1)}{(x+a)^{3}}}}$

(3) ${\displaystyle 1+x+{\dfrac {x^{2}}{1\times 2}}+{\dfrac {x^{3}}{1\times 2\times 3}}}$.

(4) (Exercises III.):

(1) (a) ${\displaystyle {\dfrac {d^{2}y}{dx^{2}}}={\dfrac {d^{3}y}{dx^{3}}}=1+x+{\frac {1}{2}}x^{2}+{\frac {1}{6}}x^{3}+\ldots }$

(b) ${\displaystyle 2a}$, ${\displaystyle 0}$.

(c) ${\displaystyle 2,0}$.

(d) ${\displaystyle 6x+6a,6}$.

​(2) ${\displaystyle -b}$, ${\displaystyle 0}$.

(3) ${\displaystyle 2}$, ${\displaystyle 0}$.

(4) ${\displaystyle 56440x^{3}-196212x^{2}-4488x+8192}$.
${\displaystyle \quad \quad \quad 169320x^{2}-392424x-4488}$.

(5) ${\displaystyle 2}$, ${\displaystyle 0}$.

(6) ${\displaystyle 371.80453x}$, ${\displaystyle 371.80453}$.

(7) ${\displaystyle {\dfrac {30}{(3x+2)^{3}}}}$, ${\displaystyle -{\dfrac {270}{(3x+2)^{4}}}}$.

(Examples, p. 41):

(1) ${\displaystyle {\dfrac {6a}{b^{2}}}x}$, ${\displaystyle {\dfrac {6a}{b^{2}}}}$.

(2) ${\displaystyle {\dfrac {3a{\sqrt {b}}}{2{\sqrt {x}}}}-{\dfrac {6b{\sqrt[{3}]{a}}}{x^{3}}}}$, ${\displaystyle {\dfrac {18b{\sqrt[{3}]{a}}}{x^{4}}}-{\dfrac {3a{\sqrt {b}}}{4{\sqrt {x^{3}}}}}}$.

(3) ${\displaystyle {\dfrac {2}{\sqrt[{3}]{\theta ^{8}}}}-{\dfrac {1.056}{\sqrt[{5}]{\theta ^{11}}}}}$, ${\displaystyle {\dfrac {2.3232}{\sqrt[{5}]{\theta ^{16}}}}-{\dfrac {16}{3{\sqrt[{3}]{\theta ^{11}}}}}}$.

(4) ${\displaystyle 810t^{4}-648t^{3}+479.52t^{2}-139.968t+26.64}$.
${\displaystyle \quad \quad \quad 3240t^{3}-1944t^{2}+959.04t-139.968}$.

(5) ${\displaystyle 12x+2}$, ${\displaystyle 12}$.

(6) ${\displaystyle 6x^{2}-9x}$, ${\displaystyle 12x-9}$.

(7) ${\displaystyle {\begin{aligned}&{\dfrac {3}{4}}\left({\dfrac {1}{\sqrt {\theta }}}+{\dfrac {1}{\sqrt {\theta ^{5}}}}\right)+{\dfrac {1}{4}}\left({\dfrac {15}{\sqrt {\theta ^{7}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right).\\&{\dfrac {3}{8}}\left({\dfrac {1}{\sqrt {\theta ^{5}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right)-{\dfrac {15}{8}}\left({\dfrac {7}{\sqrt {\theta ^{9}}}}+{\dfrac {1}{\sqrt {\theta ^{7}}}}\right).\end{aligned}}}$

---

(2) ${\displaystyle 64}$; ${\displaystyle 147.2}$; and ${\displaystyle 0.32}$ feet per second.

(3) ${\displaystyle {\dot {x}}=a-gt}$; ${\displaystyle {\ddot {x}}=-g}$.

(4) ${\displaystyle 45.1}$ feet per second.

(5) ${\displaystyle 12.4}$ feet per second per second. Yes.

(6) Angular velocity ${\displaystyle =11.2}$ radians per second; angular acceleration ${\displaystyle =9.6}$ radians per second per second.

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

(8) ${\displaystyle v={\dfrac {1}{30{\sqrt[{3}]{(t-125)^{2}}}}}}$, ${\displaystyle a=-{\dfrac {1}{45{\sqrt[{3}]{(t-125)^{5}}}}}}$.

(9) ${\displaystyle v=0.8-{\dfrac {8t}{(4+t^{2})^{2}}}}$, ${\displaystyle a={\dfrac {24t^{2}-32}{(4+t^{2})^{3}}}}$, ${\displaystyle 0.7926}$ and ${\displaystyle 0.00211}$.

(10) ${\displaystyle n=2}$, ${\displaystyle n=11}$.

---

​

(1) ${\displaystyle {\dfrac {x}{\sqrt {x^{2}+1}}}}$.

(2) ${\displaystyle {\dfrac {x}{\sqrt {x^{2}+a^{2}}}}}$.

(3) ${\displaystyle -{\dfrac {1}{2{\sqrt {(a+x)^{3}}}}}}$.

(4) ${\displaystyle {\dfrac {ax}{\sqrt {(a-x^{2})^{3}}}}}$.

(5) ${\displaystyle {\dfrac {2a^{2}-x^{2}}{x^{3}{\sqrt {x^{2}-a^{2}}}}}}$.

(6) ${\displaystyle {\dfrac {{\frac {3}{2}}x^{2}\left[{\frac {8}{9}}x\left(x^{3}+a\right)-\left(x^{4}+a\right)\right]}{(x^{4}+a)^{\frac {2}{3}}(x^{3}+a)^{\frac {3}{2}}}}}$.

(7) ${\displaystyle {\dfrac {2a\left(x-a\right)}{(x+a)^{3}}}}$.

(8) ${\displaystyle {\frac {5}{2}}y^{3}}$.

(9) ${\displaystyle {\dfrac {1}{(1-\theta ){\sqrt {1-\theta ^{2}}}}}}$.

---

(1) ${\displaystyle {\dfrac {dw}{dx}}={\dfrac {3x^{2}\left(3+3x^{3}\right)}{27\left({\frac {1}{2}}x^{3}+{\frac {1}{4}}x^{6}\right)^{3}}}}$.

(2) ${\displaystyle {\dfrac {dv}{dx}}=-{\dfrac {12x}{{\sqrt {1+{\sqrt {2}}+3x^{2}}}\left({\sqrt {3}}+4{\sqrt {1+{\sqrt {2}}+3x^{2}}}\right)^{2}}}}$.

(3) ${\displaystyle {\dfrac {du}{dx}}=-{\dfrac {x^{2}\left({\sqrt {3}}+x^{3}\right)}{\sqrt {\left[1+\left(1+{\dfrac {x^{3}}{\sqrt {3}}}\right)^{2}\right]^{3}}}}}$.

---

(2) ${\displaystyle 1.44}$.

(4) ${\displaystyle {\dfrac {dy}{dx}}=3x^{2}+3}$; and the numerical values are: ${\displaystyle 3}$, ${\displaystyle 3{\frac {3}{4}}}$, ${\displaystyle 6}$, and ${\displaystyle 15}$.

(5) ${\displaystyle \pm {\sqrt {2}}}$.

(6) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {4}{9}}{\dfrac {x}{y}}}$. Slope is zero where ${\displaystyle x=0}$; and is ${\displaystyle \mp {\dfrac {1}{3{\sqrt {2}}}}}$ where ${\displaystyle x=1}$.

(7) ${\displaystyle m=4}$, ${\displaystyle n=-3}$.

(8) Intersections at ${\displaystyle x=1}$, ${\displaystyle x=-3}$. Angles ${\displaystyle 153^{\circ }\;26'}$, ${\displaystyle 2^{\circ }\;28'}$.

(9) Intersections at ${\displaystyle x=3.57}$, ${\displaystyle x=3.50}$. Angles ${\displaystyle 16^{\circ }\;16'}$.

(10) ${\displaystyle x={\tfrac {1}{3}}}$, ${\displaystyle y=2{\tfrac {1}{3}}}$, ${\displaystyle b=-{\tfrac {5}{3}}}$.

---

​

(1) Min.: ${\displaystyle x=0}$, ${\displaystyle y=0}$; max.: ${\displaystyle x=-2}$, ${\displaystyle y=-4}$.

(2) ${\displaystyle x=a}$.

(4) ${\displaystyle 25{\sqrt {3}}}$ square inches.

(5) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {10}{x^{2}}}+{\dfrac {10}{(8-x)^{2}}}}$; ${\displaystyle x=4}$; ${\displaystyle y=5}$.

(6) Max. for ${\displaystyle x=-1}$; min. for ${\displaystyle x=1}$.

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

(8) ${\displaystyle r={\tfrac {2}{3}}R}$, ${\displaystyle r={\dfrac {R}{2}}}$, no max.

(9) ${\displaystyle r=R{\sqrt {\dfrac {2}{3}}}}$, ${\displaystyle r={\dfrac {R}{\sqrt {2}}}}$, ${\displaystyle r=0.8506R}$.

(10) At the rate of ${\displaystyle {\dfrac {8}{r}}}$ square feet per second.

(11) ${\displaystyle r={\dfrac {R{\sqrt {8}}}{3}}}$.

(12) ${\displaystyle n={\sqrt {\dfrac {NR}{r}}}}$.

---

(1) Max.: ${\displaystyle x=-2.19}$, ${\displaystyle y=24.19}$; min.: ${\displaystyle x=1.52}$, ${\displaystyle y=-1.38}$.

(2) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {b}{a}}-2cx}$; ${\displaystyle {\dfrac {d^{2}y}{dx^{2}}}=-2c}$; ${\displaystyle x={\dfrac {b}{2ac}}}$ (a maximum).

(3) (a) One maximum and two minima. (b) One maximum. (${\displaystyle x=0}$; other points unreal.)

(4) Min.: ${\displaystyle x=1.71}$, ${\displaystyle y=6.14}$.

(5) Max: ${\displaystyle x=-.5}$, ${\displaystyle y=4}$.

(6) Max.: ${\displaystyle x=1.414}$, ${\displaystyle y=1.7675}$. Min.: ${\displaystyle x=-1.414}$, ${\displaystyle y=1.7675}$.

(7) Max.: ${\displaystyle x=-3.565}$, ${\displaystyle y=2.12}$. Min.: ${\displaystyle x=+3.565}$, ${\displaystyle y=7.88}$.

(8) ${\displaystyle 0.4N}$, ${\displaystyle 0.6N}$.

(9) ${\displaystyle x={\sqrt {\dfrac {a}{c}}}}$.

(10) Speed ${\displaystyle 8.66}$ nautical miles per hour. Time taken ${\displaystyle 115.47}$ hours. Minimum cost £${\displaystyle 112}$. ${\displaystyle 12s}$.

​(11) Max. and min. for ${\displaystyle x=7.5}$, ${\displaystyle y=\pm 5.414}$. (See example no. 10, here.)

(12) Min.: ${\displaystyle x={\tfrac {1}{2}}}$, ${\displaystyle y=0.25}$; max.: ${\displaystyle x=-{\tfrac {1}{3}}}$, ${\displaystyle y=1.408}$.

---

(1) ${\displaystyle {\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}}$.

(2) ${\displaystyle {\dfrac {1}{x-1}}+{\dfrac {2}{x-2}}}$.

(3) ${\displaystyle {\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}}$.

(4) ${\displaystyle {\dfrac {5}{x-4}}-{\dfrac {4}{x-3}}}$.

(5) ${\displaystyle {\dfrac {19}{13(2x+3)}}-{\dfrac {22}{13(3x-2)}}}$.

(6) ${\displaystyle {\dfrac {2}{x-2}}+{\dfrac {4}{x-3}}-{\dfrac {5}{x-4}}}$.

(7) ${\displaystyle {\dfrac {1}{6(x-1)}}+{\dfrac {11}{15(x+2)}}+{\dfrac {1}{10(x-3)}}}$.

(8) ${\displaystyle {\dfrac {7}{9(3x+1)}}+{\dfrac {71}{63(3x-2)}}-{\dfrac {5}{7(2x+1)}}}$.

(9) ${\displaystyle {\dfrac {1}{3(x-1)}}+{\dfrac {2x+1}{3(x^{2}+x+1)}}}$.

(10) ${\displaystyle x+{\dfrac {2}{3(x+1)}}+{\dfrac {1-2x}{3(x^{2}-x+1)}}}$.

(11) ${\displaystyle {\dfrac {3}{(x+1)}}+{\dfrac {2x+1}{x^{2}+x+1}}}$.

(12) ${\displaystyle {\dfrac {1}{x-1}}-{\dfrac {1}{x-2}}+{\dfrac {2}{(x-2)^{2}}}}$.

(13) ${\displaystyle {\dfrac {1}{4(x-1)}}-{\dfrac {1}{4(x+1)}}+{\dfrac {1}{2(x+1)^{2}}}}$.

(14) ${\displaystyle {\dfrac {4}{9(x-1)}}-{\dfrac {4}{9(x+2)}}-{\dfrac {1}{3(x+2)^{2}}}}$.

(15) ${\displaystyle {\dfrac {1}{x+2}}-{\dfrac {x-1}{x^{2}+x+1}}-{\dfrac {1}{(x^{2}+x+1)^{2}}}}$.

(16) ${\displaystyle {\dfrac {5}{x+4}}-{\dfrac {32}{(x+4)^{2}}}+{\dfrac {36}{(x+4)^{3}}}}$.

(17) ${\displaystyle {\dfrac {7}{9(3x-2)^{2}}}+{\dfrac {55}{9(3x-2)^{3}}}+{\dfrac {73}{9(3x-2)^{4}}}}$.

(18) ${\displaystyle {\dfrac {1}{6(x-2)}}+{\dfrac {1}{3(x-2)^{2}}}-{\dfrac {x}{6(x^{2}+2x+4)}}}$.

---

​

(1) ${\displaystyle ab(\epsilon ^{ax}+\epsilon ^{-ax})}$.

(2) ${\displaystyle 2at+{\frac {2}{t}}}$.

(3) ${\displaystyle \log _{\epsilon }n}$.

(5) ${\displaystyle npv^{n-1}}$.

(6) ${\displaystyle {\frac {n}{x}}}$.

(7) ${\displaystyle {\frac {3\epsilon ^{-{\frac {x}{x-1}}}}{(x-1)^{2}}}}$.

(8) ${\displaystyle 6x\epsilon ^{-5x}-5(3x^{2}+1)\epsilon ^{-5x}}$.

(9) ${\displaystyle {\frac {ax^{a-1}}{x^{a}+a}}}$.

(10) ${\displaystyle \left({\frac {6x}{3x^{2}-1}}+{\frac {1}{2\left({\sqrt {x}}+x\right)}}\right)\left(3x^{2}-1\right)\left({\sqrt {x}}+1\right)}$.

(11) ${\displaystyle {\frac {1-\log _{\epsilon }\left(x+3\right)}{\left(x+3\right)^{2}}}}$.

(12) ${\displaystyle a^{x}\left(ax^{a-1}+x^{a}\log _{\epsilon }a\right)}$.

(14) Min.: ${\displaystyle y=0.7}$ for ${\displaystyle x=0.694}$.

(15) ${\displaystyle {\frac {1+x}{x}}}$.

(16) ${\displaystyle {\frac {3}{x}}(\log _{\epsilon }ax)^{2}}$.

---

(1) Let ${\displaystyle {\frac {t}{T}}=x}$ (∴ ${\displaystyle t=8x}$), and use the Table on page 159.

(2) ${\displaystyle T=34.627}$; ${\displaystyle 159.46}$ minutes.

(3) Take ${\displaystyle 2t=x}$; and use the Table on page 159.

(5) (a) ${\displaystyle x^{x}\left(1+\log _{\epsilon }x\right)}$; (b) ${\displaystyle 2x(\epsilon ^{x})^{x}}$; (c) ${\displaystyle \epsilon ^{x^{x}}\times x^{x}\left(1+\log _{\epsilon }x\right)}$.

(6) ${\displaystyle 0.14}$ second.

(7) (a) ${\displaystyle 1.642}$; (b) ${\displaystyle 15.58}$.

(8) ${\displaystyle \mu =0.00037}$, ${\displaystyle 31^{m}{\frac {1}{4}}}$.

(9) ${\displaystyle i}$ is ${\displaystyle 63.4\%}$ of ${\displaystyle i_{0}}$, ${\displaystyle 220}$ kilometres.

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

(11) Min. for ${\displaystyle x={\frac {1}{\epsilon }}}$.

(12) Max. for ${\displaystyle x=\epsilon }$.

(13) Min. for ${\displaystyle x=\log _{\epsilon }a}$.

---

​

(1) (i) ${\displaystyle {\dfrac {dy}{d\theta }}=A\cos \left(\theta -{\dfrac {\pi }{2}}\right)}$;

(ii) ${\displaystyle {\dfrac {dy}{d\theta }}=2\sin \theta \cos \theta =\sin 2\theta }$ and ${\displaystyle {\dfrac {dy}{d\theta }}=2\cos 2\theta }$;

(iii) ${\displaystyle {\dfrac {dy}{d\theta }}=3\sin ^{2}\theta \cos \theta }$ and ${\displaystyle {\dfrac {dy}{d\theta }}=3\cos 3\theta }$.

(2) ${\displaystyle \theta =45^{\circ }}$ or ${\displaystyle {\dfrac {\pi }{4}}}$ radians.

(3) ${\displaystyle {\dfrac {dy}{dt}}=-n\sin 2\pi nt}$.

(4) ${\displaystyle a^{x}\log _{\epsilon }a\cos a^{x}}$.

(5) ${\displaystyle {\dfrac {\cos x}{\sin x}}={\text{cotan}}\;x}$.

(6) ${\displaystyle 18.2\cos \left(x+26^{\circ }\right)}$.

(7) The slop is ${\displaystyle {\dfrac {dy}{d\theta }}=100\cos \left(\theta -15^{\circ }\right)}$, which is a maximum when ${\displaystyle (\theta -15^{\circ })=0}$ or ${\displaystyle \theta =15^{\circ }}$; the value of the slope being then ${\displaystyle =100}$. When ${\displaystyle \theta =75^{\circ }}$ the slope is ${\displaystyle 100\cos(75^{\circ }-15^{\circ })=100\cos 60^{\circ }=100\times {\frac {1}{2}}=50}$.

(8) ${\displaystyle \cos \theta \sin 2\theta +2\cos 2\theta \sin \theta =2\sin \theta \left(\cos ^{2}\theta +\cos 2\theta \right)}$

${\displaystyle =2\sin \theta \left(3\cos ^{2}\theta -1\right)}$.

(9) ${\displaystyle amn\theta ^{n-1}\tan ^{m-1}\left(\theta ^{n}\right)\sec ^{2}\theta ^{n}}$.

(10) ${\displaystyle \epsilon ^{x}\left(\sin ^{2}x+\sin 2x\right)}$; ${\displaystyle \epsilon ^{x}\left(\sin ^{2}x+2\sin 2x+2\cos 2x\right)}$.

(11) (i) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {ab}{\left(x+b\right)^{2}}}}$;

(ii) ${\displaystyle {\dfrac {a}{b}}\epsilon ^{-{\frac {x}{b}}}}$;

(iii) ${\displaystyle {\dfrac {1}{90^{\circ }}}\times {\dfrac {ab}{\left(b^{2}+x^{2}\right)}}}$.

(12) (i) ${\displaystyle {\dfrac {dy}{dx}}=\sec x\tan x}$;

(ii) ${\displaystyle {\dfrac {dy}{dx}}=-{\dfrac {1}{\sqrt {1-x^{2}}}}}$;

(iii) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {1}{1+x^{2}}}}$;

(iv) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {1}{x{\sqrt {x^{2}-1}}}}}$;

(v) ${\displaystyle {\dfrac {dy}{dx}}={\dfrac {{\sqrt {3\sec x}}\left(3\sec ^{2}x-1\right)}{2}}}$.

(13) ${\displaystyle {\dfrac {dy}{d\theta }}=4.6\left(2\theta +3\right)^{1.3}\cos \left(2\theta +3\right)^{2.3}}$.

​(14) ${\displaystyle {\dfrac {dy}{d\theta }}=3\theta ^{2}+3\cos \left(\theta +3\right)-\log _{\epsilon }3\left(\cos \theta \times 3^{\sin \theta }+3\theta \right)}$.

(15) ${\displaystyle \theta =\cot \theta ;\theta =\pm 0.86}$; is max. for ${\displaystyle +\theta }$, min. for ${\displaystyle -\theta }$.

---

(1) ${\displaystyle x^{3}-6x^{2}y-2y^{2}}$; ${\displaystyle {\frac {1}{3}}-2x^{3}-4xy}$.

(2) ${\displaystyle 2xyz+y^{2}z+z^{2}y+2xy^{2}z^{2}}$;

${\displaystyle 2xyz+x^{2}z+xz^{2}+2x^{2}yz^{2}}$;

${\displaystyle 2xyz+x^{2}y+xy^{2}+2x^{2}y^{2}z}$.

(3) ${\displaystyle {\dfrac {1}{r}}\{\left(x-a\right)+\left(y-b\right)+\left(z-c\right)\}={\dfrac {\left(x+y+z\right)-\left(a+b+c\right)}{r}}}$; ${\displaystyle {\dfrac {3}{r}}}$.

(4) ${\displaystyle dy=vu^{v-1}\,du+u^{v}\log _{\epsilon }u\,dv}$.

(5) ${\displaystyle dy=3\sin vu^{2}\,du+u^{3}\cos v\,dv}$,

${\displaystyle dy=u\sin x^{u-1}\cos x\,dx+(\sin x)^{u}\log _{\epsilon }\sin xdu}$,

${\displaystyle dy={\dfrac {1}{v}}\,{\dfrac {1}{u}}\,du-\log _{\epsilon }u{\dfrac {1}{v^{2}}}\,dv}$.

(7) Minimum for ${\displaystyle x=y=-{\tfrac {1}{2}}}$.

(8) (a) Length ${\displaystyle 2}$ feet, width = depth = ${\displaystyle 1}$ foot, vol. = ${\displaystyle 2}$ cubic feet.

(b) Radius = ${\displaystyle 2\pi }$ feet = ${\displaystyle 7.46}$ in., length = ${\displaystyle 2}$ feet, vol. = ${\displaystyle 2.54}$.

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

(10) Minimum for ${\displaystyle x=y=1}$.

(11) Min.: ${\displaystyle x={\tfrac {1}{2}}}$ and ${\displaystyle y=2}$.

(12) Angle at apex ${\displaystyle =90^{\circ }}$; equal sides = length =${\displaystyle {\sqrt[{3}]{2V}}}$.

---

(1) ${\displaystyle 1{\tfrac {1}{3}}}$.

(2) ${\displaystyle 0.6344}$.

(3) ${\displaystyle 0.2624}$.

(4) (a) ${\displaystyle y={\tfrac {1}{8}}x^{2}+C}$;

(b) ${\displaystyle y=\sin x+C}$.

(5) ${\displaystyle y=x^{2}+3x+C}$.

---

​

(1) ${\displaystyle {\dfrac {4{\sqrt {a}}x^{\frac {3}{2}}}{3}}+C}$.

(2) ${\displaystyle -{\dfrac {1}{x^{3}}}+C}$.

(3) ${\displaystyle {\dfrac {x^{4}}{4a}}+C}$.

(4) ${\displaystyle {\tfrac {1}{3}}x^{3}+ax+C}$.

(5) ${\displaystyle -2x^{-{\frac {5}{2}}}+C}$.

(6) ${\displaystyle x^{4}+x^{3}+x^{2}+x+C}$.

(7) ${\displaystyle {\dfrac {ax^{2}}{4}}+{\dfrac {bx^{3}}{9}}+{\dfrac {cx^{4}}{16}}+C}$.

(8) ${\displaystyle {\dfrac {x^{2}+a}{x+a}}=x-a+{\dfrac {a^{2}+a}{x+a}}}$ by division. Therefore the answer is ${\displaystyle {\dfrac {x^{2}}{2}}-ax+(a^{2}+a)\log _{\epsilon }(x+a)+C}$. (See pages 199 and 201.)

(9) ${\displaystyle {\dfrac {x^{4}}{4}}+3x^{3}+{\dfrac {27}{2}}x^{2}+27x+C}$.

(10) ${\displaystyle {\dfrac {x^{3}}{3}}+{\dfrac {2-a}{2}}x^{2}-2ax+C}$.

(11) ${\displaystyle a^{2}(2x^{\frac {3}{2}}+{\tfrac {9}{4}}x^{\frac {4}{3}})+C}$.

(12) ${\displaystyle -{\tfrac {1}{3}}\cos \theta -{\tfrac {1}{6}}\theta +C}$.

(13) ${\displaystyle {\dfrac {\theta }{2}}+{\dfrac {\sin 2a\theta }{4a}}+C}$.

(14) ${\displaystyle {\dfrac {\theta }{2}}-{\dfrac {\sin 2\theta }{4}}+C}$.

(15) ${\displaystyle {\dfrac {\theta }{2}}-{\dfrac {\sin 2a\theta }{4a}}+C}$.

(16) ${\displaystyle {\tfrac {1}{3}}\epsilon ^{3x}}$.

(17) ${\displaystyle \log(1+x)+C}$.

(18) ${\displaystyle -\log _{\epsilon }(1-x)+C}$.

---

(1) Area ${\displaystyle =60}$; mean ordinate ${\displaystyle =10}$.

(2) Area ${\displaystyle ={\frac {2}{3}}}$ of ${\displaystyle a\times 2a{\sqrt {a}}}$.

(3) Area ${\displaystyle =2}$; mean ordinate ${\displaystyle ={\dfrac {2}{\pi }}=0.637}$.

(4) Area ${\displaystyle =1.57}$; mean ordinate ${\displaystyle =0.5}$.

(5) ${\displaystyle 0.572}$, ${\displaystyle 0.0476}$.

(6) Volume ${\displaystyle =\pi r^{2}{\dfrac {h}{3}}}$.

(7) ${\displaystyle 1.25}$.

(8) ${\displaystyle 79.4}$.

(9) Volume ${\displaystyle =4.9348}$; area of surface ${\displaystyle =12.57}$ (from ${\displaystyle 0}$ to ${\displaystyle \pi }$).

(10) ${\displaystyle a\log _{\epsilon }a}$, ${\displaystyle {\dfrac {a}{a-1}}\log _{\epsilon }a}$.

(12) Arithmetical mean ${\displaystyle =9.5}$; quadratic mean ${\displaystyle =10.85}$.

​(13) Quadratic mean ${\displaystyle ={\tfrac {1}{\sqrt {2}}}{\sqrt {A_{1}^{2}+A_{3}^{2}}}}$; arithmetical mean ${\displaystyle =0}$. 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 ${\displaystyle \sin ^{2}x}$ we write

For ${\displaystyle 2\sin x\sin 3x}$ we write ${\displaystyle \cos 2x-\cos 4x}$; and, for ${\displaystyle \sin ^{2}3x}$,

Making these substitutions, and integrating, we get (see p. 202)

At the lower limit the substitution of ${\displaystyle 0}$ for ${\displaystyle x}$ causes all this to vanish, whilst at the upper limit the substitution of ${\displaystyle 2\pi }$ for ${\displaystyle x}$ gives ${\displaystyle A_{1}^{2}\pi +A_{3}^{2}\pi }$. And hence the answer follows.

(14) Area is ${\displaystyle 62.6}$ square units. Mean ordinate is ${\displaystyle 10.42}$.

(16) ${\displaystyle 436.3}$. (This solid is pear shaped.)

---

(1) ${\displaystyle {\dfrac {x{\sqrt {a^{2}-x^{2}}}}{2}}+{\dfrac {a^{2}}{2}}\sin ^{-1}{\dfrac {x}{a}}+C}$.

(2) ${\displaystyle {\dfrac {x^{2}}{2}}(\log _{\epsilon }x-{\tfrac {1}{2}})+C}$.

(3) ${\displaystyle {\dfrac {x^{a+1}}{a+1}}\left(\log _{\epsilon }x-{\dfrac {1}{a+1}}\right)+C}$.

(4) ${\displaystyle \sin \epsilon ^{x}+C}$.

(5) ${\displaystyle \sin(\log _{\epsilon }x)+C}$.

(6) ${\displaystyle \epsilon ^{x}(x^{2}-2x+2)+C}$.

​(7) ${\displaystyle {\dfrac {1}{a+1}}(\log _{\epsilon }x)^{a+1}+C}$.

(8) ${\displaystyle \log _{\epsilon }(\log _{\epsilon }x)+C}$.

(9) ${\displaystyle 2\log _{\epsilon }(x-1)+3\log _{\epsilon }(x+2)+C}$.

(10) ${\displaystyle {\frac {1}{2}}\log _{\epsilon }(x-1)+{\frac {1}{5}}\log _{\epsilon }(x-2)+{\frac {3}{10}}\log _{\epsilon }(x+3)+C}$.

(11) ${\displaystyle {\dfrac {b}{2a}}\log _{\epsilon }{\dfrac {x-a}{x+a}}+C}$.

(12) ${\displaystyle \log _{\epsilon }{\dfrac {x^{2}-1}{x^{2}+1}}+C}$.

(13) ${\displaystyle {\frac {1}{4}}\log _{\epsilon }{\dfrac {1+x}{1-x}}+{\frac {1}{2}}\arctan x+C}$.

(14) ${\displaystyle {\dfrac {1}{\sqrt {a}}}\log _{\epsilon }{\dfrac {{\sqrt {a}}-{\sqrt {a-bx^{2}}}}{x{\sqrt {a}}}}}$. (Let ${\displaystyle {\dfrac {1}{x}}=v}$; then, in the result, let ${\displaystyle {\sqrt {v^{2}-{\dfrac {b}{a}}}}=v-u}$.)

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.