# Elements of the Differential and Integral Calculus/Chapter I

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$\scriptstyle {=2\pi r}$.
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. . . . | $\scriptstyle {-}$ | $\scriptstyle {+}$ | $\scriptstyle {-}$ | $\scriptstyle {-}$ | $\scriptstyle {+}$ | $\scriptstyle {-}$ |
| 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 |
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 |