Elements of the Differential and Integral Calculus/Chapter I

by William Anthony Granville

 

DIFFERENTIAL CALCULUS

 

CHAPTER I

 

COLLECTION OF FORMULAS

 

1. Formulas for reference. For the convenience of the student we give the following list of elementary formulas from Algebra, Geometry, Trigonometry, and Analytic Geometry.

1. Binomial Theorem ($\scriptscriptstyle {n}$ being a positive integer):

${\begin{aligned}\scriptstyle {(a+b)^{n}=a^{n}+na^{n-1}b}&\scriptstyle {+{\frac {n(n-1)}{\begin{array}{|c}\scriptscriptstyle {2}\\\hline \end{array}}}a^{n-2}b^{2}+{\frac {n(n-1)(n-2)}{\begin{array}{|c}\scriptscriptstyle {3}\\\hline \end{array}}}a^{n-3}b^{3}+\cdots }\\&\scriptstyle {+{\frac {n(n-1)(n-2)\cdots (n-r+2)}{\begin{array}{|c}\scriptscriptstyle {r-1}\\\hline \end{array}}}a^{n-r+1}b^{r-1}+\cdots .}\end{aligned}}$

2. $\scriptstyle {n!={\begin{array}{|c}\scriptscriptstyle {n}\\\hline \end{array}}=1\cdot 2\cdot 3\cdot 4\cdots (n-1)n.}$

3. In the quadratic equation $\scriptstyle {ax^{2}+bx+c=0}$,

when $\scriptstyle {b^{2}-4ac>0}$, the roots are real and unequal;
when $\scriptstyle {b^{2}-4ac=0}$, the roots are real and equal;
when $\scriptstyle {b^{2}-4ac<0}$, the roots are imaginary.

4. When a quadratic equation is reduced to the form $\scriptstyle {x^{2}+px+q=0}$,

$\scriptstyle {p=}$sum of roots with sign changed, and $\scriptstyle {q=}$product of roots.

5. In an arithmetical series,

$\scriptstyle {l=a+(n-1)d;\ s={\frac {n}{2}}(a+l)={\frac {n}{2}}[2a+(n-1)d]}$.

6. In a geometrical series,

$\scriptstyle {l=ar^{n-1};\ s={\frac {rl-a}{r-1}}={\frac {a(r^{n}-1}{r-1}}}$.

7. $\scriptstyle {\log ab=\log a+\log b}$.

8. $\scriptstyle {\log {\frac {a}{b}}=\log a-\log b}$.

9. $\scriptstyle {\log a^{n}=n\log a}$.

10. $\scriptstyle {\log {\sqrt[{n}]{a}}={\frac {1}{n}}\log a}$.

11. $\scriptstyle {\log 1=0}$.

12. $\scriptstyle {\log _{a}a=1}$.

13. $\scriptstyle {\log {\frac {1}{a}}=-\log a}$.

14. Circumference of circle$\scriptstyle {=2\pi r}$.^[1]

15. Area of circle$\scriptstyle {=\pi r^{2}}$.

16. Volume of prism$\scriptstyle {=Ba}$.

17. Volume of pyramid$\scriptstyle {={\frac {1}{3}}Ba}$.

18. Volume of right circular cylinder$\scriptstyle {=\pi r^{2}a}$.

19. Lateral surface of right circular cylinder$\scriptstyle {=2\pi ra}$.

20. Total surface of right circular cylinder$\scriptstyle {=2\pi r(r+a)}$.

21. Volume of right circular cone$\scriptstyle {={\frac {1}{3}}\pi r^{3}a}$.

22. Lateral surface of right circular cone$\scriptstyle {=\pi rs}$.

23. Total surface of right circular cone$\scriptstyle {=\pi r(r+s)}$.

24. Volume of sphere$\scriptstyle {={\frac {4}{3}}\pi r^{3}}$.

25. Surface of sphere$\scriptstyle {=4\pi r^{2}}$.

26. $\scriptstyle {\sin x={\frac {1}{\csc x}};\ \cos x={\frac {1}{\sec x}};\ \tan x={\frac {1}{\cot x}}}$.

27. $\scriptstyle {\tan x={\frac {\sin x}{\cos x}};\ \cot x={\frac {\cos x}{\sin x}}}$.

28. $\scriptstyle {\sin ^{2}x+\cos ^{2}x=1;\ 1+\tan ^{2}x=\sec ^{2}x;\ 1+\cot ^{2}x=\csc ^{2}x}$.

29. ${\begin{array}{l}\scriptstyle {\sin x=\cos \left({\frac {\pi }{2}}-x\right);}\\\scriptstyle {\cos x=\sin \left({\frac {\pi }{2}}-x\right);}\\\scriptstyle {\tan x=\cot \left({\frac {\pi }{2}}-x\right).}\end{array}}$

30. ${\begin{array}{l}\scriptstyle {\sin(\pi -x)=\sin x;}\\\scriptstyle {\cos(\pi -x)=-\cos x;}\\\scriptstyle {\tan(\pi -x)=-\tan(x).}\end{array}}$

31. $\scriptstyle {\sin(x+y)=\sin x\cos y+\cos x\sin y}$.

32. $\scriptstyle {\sin(x-y)=\sin x\cos y-\cos x\sin y}$.

33. $\scriptstyle {\cos(x\pm y)=\cos x\cos y\mp \sin x\sin y}$.

34. $\scriptstyle {\tan(x+y)={\frac {\tan x+\tan y}{1-\tan x\tan y}}}$.

35. $\scriptstyle {\tan(x-y)={\frac {\tan x-\tan y}{1+\tan x\tan y}}}$.

36. $\scriptstyle {\sin 2x=2\sin x\cos x;\ \cos 2x=\cos ^{2}x=\sin ^{2}x;\ \tan 2x={\frac {2\tan x}{1-\tan ^{2}x}}}$.

37. $\scriptstyle {\sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}};\ \cos x=\cos ^{2}{\frac {x}{2}}-\sin ^{2}{\frac {x}{2}};\ \tan x={\frac {2\tan {\frac {1}{2}}x}{1-\tan ^{2}{\frac {1}{2}}x}}}$.

38. $\scriptstyle {\cos ^{2}x={\frac {1}{2}}+{\frac {1}{2}}\cos 2x;\ \sin ^{2}x={\frac {1}{2}}-{\frac {1}{2}}\cos 2x}$.

39. $\scriptstyle {1+\cos x=2\cos ^{2}{\frac {x}{2}};\ 1-\cos x=2\sin ^{2}{\frac {x}{2}}}$.

40. $\scriptstyle {\sin {\frac {x}{2}}=\pm {\sqrt {\frac {1-\cos x}{2}}};\ \cos {\frac {x}{2}}=\pm {\sqrt {\frac {1+\cos x}{2}}};\ \tan {\frac {x}{2}}=\pm {\sqrt {\frac {1-\cos x}{1+\cos x}}}}$.

41. $\scriptstyle {\sin x+\sin y=2\sin {\frac {1}{2}}(x+y)\cos {\frac {1}{2}}(x-y)}$.

42. $\scriptstyle {\sin x-\sin y=2\cos {\frac {1}{2}}(x+y)\sin {\frac {1}{2}}(x-y)}$.

43. $\scriptstyle {\cos x+\cos y=2\cos {\frac {1}{2}}(x+y)\cos {\frac {1}{2}}(x-y)}$.

44. $\scriptstyle {\cos x-\cos y=-2\sin {\frac {1}{2}}(x+y)\sin {\frac {1}{2}}(x-y)}$.

45. $\scriptstyle {{\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}};}$ Law of Sines.

46. $\scriptstyle {a^{2}=b^{2}+c^{2}-2bc\cos A;}$ Law of Cosines.

47. $\scriptstyle {d={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}};}$ distance between points $\scriptstyle {(x_{1},y_{1})}$ and $\scriptstyle {(x_{2},y_{2})}$.

48. $\scriptstyle {d={\frac {Ax_{1}+By_{1}+C}{\pm {\sqrt {A^{2}+B^{2}}}}};}$ distance from line $\scriptstyle {Ax+By+C=0}$ to $\scriptstyle {(x_{1},y_{1})}$.

49. $\scriptstyle {x={\frac {x_{1}+x_{2}}{2}},~y={\frac {y_{1}+y_{2}}{2}}}$; coördinates of middle point.

50. $\scriptstyle {x=x_{0}+x',~y=y_{0}+y'}$; transforming to new origin $\scriptstyle {(x_{0},y_{0})}$.

51. $\scriptstyle {x=x'cos\theta -y'sin\theta ,~y=x'sin\theta +y'cos\theta }$; transforming to new axes making the angle $\scriptstyle {\theta }$ with old.

52. $\scriptstyle {x=\rho \cos \theta ,~y=\rho \sin \theta }$; transforming from rectangular to polar coördinates.

53. $\scriptstyle {\rho ={\sqrt {x^{2}+y^{2}}},~\theta =\operatorname {arc~tan} {\frac {y}{x}}}$; transforming from polar to rectangular coördinates.

54. Different forms of equation of a straight line:

(a) $\scriptstyle {{\frac {y-y_{1}}{x-x_{1}}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$, two-point form;
(b) $\scriptstyle {{\frac {x}{a}}+{\frac {y}{b}}=1}$, intercept form;
(c) $\scriptstyle {y-y_{1}=m(x-x_{1})}$, slope-point form;
(d) $\scriptstyle {y=mx+b}$, slope-intercept form;
(e) $\scriptstyle {x\cos \alpha +y\sin \alpha =p}$, normal form;
(f) $\scriptstyle {Ax+By+C=0}$, general form.

55. $\scriptstyle {\tan \theta ={\frac {m_{1}-m_{2}}{1+m_{1}m_{2}}}}$, angle between two lines whose slopes are $\scriptstyle {m_{1}}$ and $\scriptstyle {m_{2}}$.
$\scriptstyle {m_{1}=m_{2}}$ when lines are parallel,
and$\scriptstyle {m_{1}=-{\frac {1}{m_{2}}}}$ when lines are perpendicular.

56. $\scriptstyle {(x-\alpha )^{2}+(y-\beta )^{2}=r^{2}}$, equation of circle with center $\scriptstyle {(\alpha ,\beta )}$ and radius $\scriptstyle {r}$.

2. Greek alphabet.

Letters	Names	Letters	Names	Letters	Names
$\scriptstyle {\mathrm {A} \quad \alpha }$	Alpha	$\scriptstyle {\mathrm {I} \quad \iota }$	Iota	$\scriptstyle {\mathrm {P} \quad \rho }$	Rho
$\scriptstyle {\mathrm {B} \quad \beta }$	Beta	$\scriptstyle {\mathrm {K} \quad \kappa }$	Kappa	$\scriptstyle {\Sigma \quad \sigma }$	Sigma
$\scriptstyle {\Gamma \quad \gamma }$	Gamma	$\scriptstyle {\Lambda \quad \lambda }$	Lambda	$\scriptstyle {\mathrm {T} \quad \tau }$	Tau
$\scriptstyle {\Delta \quad \delta }$	Delta	$\scriptstyle {\mathrm {M} \quad \mu }$	Mu	$\scriptstyle {\Upsilon \quad \upsilon }$	Upsilon
$\scriptstyle {\mathrm {E} \quad \epsilon }$	Epsilon	$\scriptstyle {\mathrm {N} \quad \nu }$	Nu	$\scriptstyle {\Phi \quad \phi }$	Phi
$\scriptstyle {\mathrm {Z} \quad \zeta }$	Zeta	$\scriptstyle {\Xi \quad \xi }$	Xi	$\scriptstyle {\mathrm {X} \quad \chi }$	Chi
$\scriptstyle {\mathrm {H} \quad \eta }$	Eta	$\scriptstyle {\mathrm {O} \quad \mathrm {o} }$	Omicron	$\scriptstyle {\Psi \quad \psi }$	Psi
$\scriptstyle {\Theta \quad \theta }$	Theta	$\scriptstyle {\Pi \quad \pi }$	Pi	$\scriptstyle {\Omega \quad \omega }$	Omega

3. Rules for signs of the trigonometric functions.

Quadrant	Sin	Cos	Tan	Cot	Sec	Csc
First. . . .	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$
Second. . . .	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$
Third. . . .	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$
Fourth. . . .	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {-}$

4. Natural Values of the trigonometric functions.

Angle in Radians	Angle in Degrees	Sin	Cos	Tan	Cot	Sec	Csc
$\scriptstyle {0}$	0°	$\scriptstyle {0}$	$\scriptstyle {1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {1}$	$\scriptstyle {\infty }$
$\scriptstyle {\frac {\pi }{2}}$	90°	$\scriptstyle {1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {1}$
$\scriptstyle {\pi }$	180°	$\scriptstyle {0}$	$\scriptstyle {-1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {-1}$	$\scriptstyle {\infty }$
$\scriptstyle {\frac {3\pi }{2}}$	270°	$\scriptstyle {-1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {-1}$
$\scriptstyle {2\pi }$	360°	$\scriptstyle {0}$	$\scriptstyle {1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {1}$	$\scriptstyle {\infty }$

 

Angle in Radians	Angle in Degrees	Sin	Cos	Tan	Cot	Sec	Csc
$\scriptstyle {0}$	0°	$\scriptstyle {0}$	$\scriptstyle {1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {1}$	$\scriptstyle {\infty }$
$\scriptstyle {\frac {\pi }{6}}$	30°	$\scriptstyle {\frac {1}{2}}$	$\scriptstyle {\frac {\sqrt {3}}{2}}$	$\scriptstyle {\frac {\sqrt {3}}{3}}$	$\scriptstyle {\sqrt {3}}$	$\scriptstyle {\frac {2{\sqrt {3}}}{3}}$	$\scriptstyle {2}$
$\scriptstyle {\frac {\pi }{4}}$	45°	$\scriptstyle {\frac {\sqrt {2}}{2}}$	$\scriptstyle {\frac {\sqrt {2}}{2}}$	$\scriptstyle {1}$	$\scriptstyle {1}$	$\scriptstyle {\sqrt {2}}$	$\scriptstyle {\sqrt {2}}$
$\scriptstyle {\frac {\pi }{3}}$	60°	$\scriptstyle {\frac {\sqrt {3}}{2}}$	$\scriptstyle {\frac {1}{2}}$	$\scriptstyle {\sqrt {3}}$	$\scriptstyle {\frac {\sqrt {3}}{3}}$	$\scriptstyle {2}$	$\scriptstyle {\frac {2{\sqrt {3}}}{3}}$
$\scriptstyle {\frac {\pi }{2}}$	90°	$\scriptstyle {1}$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {0}$	$\scriptstyle {\infty }$	$\scriptstyle {1}$

 

Angle in Radians	Angle in Degrees	Sin	Cos	Tan	Cot
.0000	0°	.0000	1.0000	.0000	$\scriptstyle {\infty }$	90°	1.5708
.0175	1°	.0175	.9998	.0175	57.290	89°	1.5533
.0349	2°	.0349	.9994	.0349	28.636	88°	1.5359
.0524	3°	.0523	.9986	.0524	19.081	87°	1.5184
.0698	4°	.0698	.9976	.0699	14.300	86°	1.5010
.0873	5°	.0872	.9962	.0875	11.430	85°	1.4835
.1745	10°	.1736	.9848	.1763	5.671	80°	1.3963
.2618	15°	.2588	.9659	.2679	3.732	75°	1.3090
.3491	20°	.3420	.9397	.3640	2.747	70°	1.2217
.4363	25°	.4226	.9063	.4663	2.145	65°	1.1345
.5236	30°	.5000	.8660	.5774	1.732	60°	1.0472
.6109	35°	.5736	.8192	.7002	1.428	55°	.9599
.6981	40°	.6428	.7660	.8391	1.192	50°	.8727
.7854	45°	.7071	.7071	1.0000	1.000	45°	.7854
		Cos	Sin	Cot	Tan	Angle in Degrees	Angle in Radians

5. Logarithms of numbers and trigonometric functions.

Table of Mantissas of the Common Logarithms of Numbers

No.	0	1	2	3	4	5	6	7	8	9
1	0000	0414	0792	1139	1461	1761	2041	2304	2553	2788
2	3010	3222	3424	3617	3802	3979	4150	4314	4472	4624
3	4771	4914	5051	5185	5315	5441	5563	5682	5798	5911
4	6021	6128	6232	6335	6435	6532	6628	6721	6812	6902
5	6990	7076	7160	7243	7324	7404	7482	7559	7634	7709
6	7782	7853	7924	7993	8062	8129	8195	8261	8325	8388
7	8451	8513	8573	8633	8692	8751	8808	8865	8921	8976
8	9031	9085	9138	9191	9243	9294	9345	9395	9445	9494
9	9542	9590	9638	9685	9731	9777	9823	9868	9912	9956
10	0000	0043	0086	0128	0170	0212	0253	0294	0334	0374
11	0414	0453	0492	0531	0569	0607	0645	0682	0719	0755
12	0792	0828	0864	0899	0934	0969	1004	1038	1072	1106
13	1139	1173	1206	1239	1271	1303	1335	1367	1339	1430
14	1461	1492	1523	1553	1584	1614	1644	1673	1703	1732
15	1761	1790	1818	1847	1875	1903	1931	1959	1987	2014
16	2041	2068	2095	2122	2148	2175	2201	2227	2253	2279
17	2304	2330	2355	2380	2405	2430	2455	2480	2504	2529
18	2553	2577	2601	2625	2648	2672	2695	2718	2742	2765
19	2788	2810	2833	2856	2878	2900	2923	2945	2967	2989

 

Table of Logarithms of the Trigonometric Functions

Angle in Radians	Angle in Degrees	log sin	log cos	log tan	log cot
.0000	0°	· · · ·	0.000	· · · ·	· · · ·	90°	1.5708
.0175	1°	8.2419	9.9999	8.2419	1.7581	89°	1.5533
.0349	2°	8.5428	9.9997	8.5431	1.4569	88°	1.5359
.0524	3°	8.7188	9.9994	8.7194	1.2806	87°	1.5184
.0698	4°	8.8436	9.9989	8.8446	1.1554	86°	1.5010
.0873	5°	8.9403	9.9983	8.9420	1.0580	85°	1.4835
.1745	10°	9.2397	9.9934	9.2463	0.7537	80°	1.3963
.2618	15°	9.4130	9.9849	9.4281	0.5719	75°	1.3090
.3491	20°	9.5341	9.9730	9.5611	0.4389	70°	1.2217
.4363	25°	9.6259	9.9573	9.6687	0.3313	65°	1.1345
.5236	30°	9.6990	9.9375	9.7614	0.2386	60°	1.0472
.6109	35°	9.7586	9.9134	9.8452	0.1548	55°	0.9599
.6981	40°	9.8081	9.8843	9.9238	0.0762	50°	0.8727
.7854	45°	9.8495	9.8495	0.0000	0.0000	45°	0.7854
		log cos	log sin	log cot	log tan	Angle in Degrees	Angle in Radians

1. In formulas 14–25, $\scriptstyle {r}$ denotes radius, $\scriptstyle {a}$ altitude, $\scriptstyle {B}$ area of base, and $\scriptstyle {s}$ slant height.