# Elements of the Differential and Integral Calculus/Chapter I

## 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 (n being a positive integer):
$\begin{smallmatrix} (a + b)^n = a^n + na^{n-1}b &+& \frac{n(n - 1)}{2!}a^{n-2}b^2 + \frac{n(n - 1)(n - 2)}{3!}a^{n-3}b^3 + \cdots \\ &+& \frac{n(n - 1)(n - 2)\cdots(n - r + 2)}{(r - 1)!}a^{n-r}b^{r-1} + \cdots \end{smallmatrix}$
2. $\begin{smallmatrix}n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n - 1)n.\end{smallmatrix}$
3. In the quadratic equation $ax^2 + bx + c = 0$,
when $b^2 - 4 ac > 0$, the roots are real and unequal;
when $b^2 - 4 ac = 0$, the roots are real and equal;
when $b^2 - 4 ac < 0$, the roots are imaginary.
4. When a quadratic equation is reduced to the form $x^2 + px + q = 0$,
$p = \text{sum of roots with sign changed, and } q = \text{product of roots}$.
5. In an arithmetical series,
$\begin{smallmatrix}l = a + (n-1)d; s = \frac{n}{2}(a + l) = \frac{n}{2}[2a + (n-1)d]\end{smallmatrix}$.
6. In a geometrical series,
$\begin{smallmatrix}l = ar^{n-1}; s= \frac{rl - a}{r - 1} = \frac{a(r^n - 1)}{r - 1}\end{smallmatrix}$.
7. $\log ab = \log a + \log b$.
8. $\begin{smallmatrix}\log \frac{a}{b} = \log a - \log b\end{smallmatrix}$.
9. $\log a^n = n \log a$.
10. $\begin{smallmatrix}\log \sqrt[n]{a} = \frac{1}{n} \log a\end{smallmatrix}$.
11. $\log 1 = 0$.
12. $\log_a a = 1$.
13. $\begin{smallmatrix}\log \frac{1}{a} = -\log a\end{smallmatrix}$.
14. Circumference of circle $\begin{smallmatrix}= 2 \pi\ r\end{smallmatrix}$.[1]
15. Area of circle $\begin{smallmatrix}= \pi\ r^2\end{smallmatrix}$.
16. Volume of prism $= Ba$.
17. Volume of pyramid $\begin{smallmatrix}= \frac{1}{3} Ba\end{smallmatrix}$.
18. Volume of right circular cylinder $\begin{smallmatrix}= \pi\ r^2a\end{smallmatrix}$.
19. Lateral surface of right circular cylinder $\begin{smallmatrix}= 2 \pi\ ra\end{smallmatrix}$.
20. Total surface of right circular cylinder $\begin{smallmatrix}= 2 \pi\ r(r + a)\end{smallmatrix}$.
21. Volume of right circular cone $\begin{smallmatrix}= 2 \pi\ r(r + a)\end{smallmatrix}$.
22. Lateral surface of right circular cone $\begin{smallmatrix}= \pi\ rs\end{smallmatrix}$.
23. Total surface of right circular cone $\begin{smallmatrix}= \pi\ r(r + s)\end{smallmatrix}$.
24. Volume of sphere $\begin{smallmatrix}= \frac{4}{3}\pi\ r^3\end{smallmatrix}$.
25. Surface of sphere $\begin{smallmatrix}= 4\pi\ r^2\end{smallmatrix}$.
26. $\begin{smallmatrix}\sin x = \frac{1}{\csc x}; \cos x = \frac{1}{\sec x}; \tan x = \frac{1}{\cot x}\end{smallmatrix}$.
27. $\begin{smallmatrix}\tan x = \frac{\sin{x}}{\cos{x}}; \cot{x} = \frac{\cos{x}}{\sin{x}}\end{smallmatrix}$.
28. $\sin^2{x} + \cos^2{x} = 1; 1 + \tan^2{x} = \sec^2{x}; 1 + \cot^2{x} = \csc^2{x}$.
29. $\sin x = \cos \left ( \frac{\pi}{2} - x \right ); \cos x = \sin \left ( \frac{\pi}{2} - x \right); \tan x = \cot \left ( \frac{\pi}{2} - x \right )$.
30. $\sin(\pi\ - x) = \sin x; \cos(\pi\ - x) = -\cos x; \tan(\pi\ - x) = -\tan x$.
31. $\sin(x + y) = \sin x \cos y + \cos x \sin y$.
32. $\sin(x - y) = \sin x \cos y - \cos x \sin y$.
33. $\cos(x \pm y) = \cos x \cos y -+ \sin x \sin y$.
34. $\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$.
35. $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$.
36. $\sin 2x = 2 \sin x \cos x; \cos 2x = \cos^2 x - \sin^2 x; \tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$
37. $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}; \cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}; \tan 2x = \frac{2 \tan \frac{1}{2} x}{1 - \tan^2 \frac{1}{2} x}$.
38. $\cos^2 x = \frac{1}{2} + \frac{1}{2} \cos 2x; \sin^2 x = \frac{1}{2} - \frac{1}{2} \cos 2x$.
39. $1 + \cos x = 2 \cos^2 \frac{x}{2}; 1 - \cos x = 2 \sin^2 \frac{x}{2}$.
40. $\sin \frac{x}{2} = \pm \sqrt{ \frac{1 - \cos x}{2} }; \cos x/2 = \pm \sqrt{ \frac{1 + \cos x}{2} }; \tan \frac{x}{2} = \pm \sqrt{ \frac{1 - \cos x}{1 + \cos x}}$.
41. $\sin x + \sin y = 2 \sin \frac{1}{2} (x + y) cos \frac{1}{2} (x - y)$.
42. $\sin x - \sin y = 2 \cos \frac{1}{2} (x + y) sin \frac{1}{2} (x - y)$.
43. $\cos x + \cos y = -2 \cos \frac{1}{2} (x + y) cos \frac{1}{2} (x - y)$.
44. $\cos x - \cos y = -2 \sin \frac{1}{2} (x + y) sin \frac{1}{2} (x - y)$.
45. $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}; \mbox{ Law of Sines}$.
46. $a^2 = b^2 + c^2 - 2 bc \cos A; \mbox{ Law of Cosines}$.
47. $d = \sqrt{ (x_1 - x_2)^2 + (y_1 - y_2)^2};$ distance between points $(x_1, y_1)$ and $(x_2, y_2)$.
48. $d = \frac{Ax_1 + By_1 + C}{\pm \sqrt{A^2 + B^2}};$ distance from line $Ax + By + C = 0$ to $(x_1, y_1)$.
49. $x = \frac{x_1 + x_2}{2} , y = \frac{y_1 + y_2}{2};$ coördinates of middle point.
50. $x = x_0 + x' , y = y_0 + y'; \mbox{ transforming to new origin } (x_0, y_0)$.
51. $x = x' \cos \theta\ - y' \sin \theta\ , y = x' \sin \theta\ + y' \cos \theta\;$ transforming to new axes making the angle theta with old.
52. $x = \rho\ \cos \theta\ , y = \rho\ \sin \theta\;$ transforming from rectangular to polar coördinates.
53. $\rho\ = \sqrt{x^2 + y^2}, \theta\ = \arctan \frac{y}{x};$ transforming from polar to rectangular coördinates.
54. Different forms of equation of a straight line:
(a) $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}, \mbox{ two-point form};$
(b) $\frac{x}{a} + \frac{y}{b} = 1, \mbox{ intercept form};$
(c) $y - y_1 = m(x - x_1), \mbox{ slope-point form};$
(d) $y = mx + b, \mbox{ slope-intercept form};$
(e) $x \cos \alpha\ + y \sin \alpha\ = p, \mbox{ normal form};$
(f) $Ax + By + C = 0, \mbox{ general form}$.
55. $\tan \theta\ = \frac{m_1 - m_2}{1 + m_1 m_2},$ angle between two lines whose slopes are $m_1$ and $m_2$.
$m_1 = m2$ when lines are parallel, and
$m_1 = -\frac{1}{m_2}$ when lines are perpendicular.
56. $(x - \alpha)^2 + {(y - \beta)}^2 = r^2,$ equation of circle with center $(\alpha, \beta)$ and radius $r$.

2. Greek alphabet.

Letters Names Letters Names Letters Names
$\Alpha$ $\alpha$ Alpha $\Iota$ $\iota$ Iota $\Rho$ $\rho$ Rho
$\Beta$ $\beta$ Beta $\Kappa$ $\kappa$ Kappa $\Sigma$ $\sigma$ Sigma
$\Gamma$ $\gamma$ Gamma $\Lambda$ $\lambda$ Lambda $\Tau$ $\tau$ Tau
$\Delta$ $\delta$ Delta $\Mu$ $\mu$ Mu $\Upsilon$ $\upsilon$ Upsilon
$\Epsilon$ $\epsilon$ Epsilon $\Nu$ $\nu$ Nu $\Phi$ $\phi$ Phi
$\Zeta$ $\zeta$ Zeta $\Xi$ $\xi$ Xi $\Chi$ $\chi$ Chi
$\Eta$ $\eta$ Eta Ο ο Omicron $\Psi$ $\psi$ Psi
$\Theta$ $\theta$ Theta $\Pi$ $\pi$ Pi $\Omega$ $\omega$ Omega

3. Rules for signs of the trigonometric functions.

Quadrant Sin Cos Tan Cot Sec Csc
First + + + + + +
Second + +
Third + +
Fourth + +

4. Natural values of the trigonometric functions.

Angle in
Angle in
Degrees
Sin Cos Tan Cot Sec Csc
0 0 1 0 $\infty$ 1 $\infty$
$\frac{\pi}{2}$ 90° 1 0 $\infty$ 0 $\infty$ 1
$\pi$ 180° 0 -1 0 $\infty$ -1 $\infty$
$\frac{3\pi}{2}$ 270° -1 0 $\infty$ 0 $\infty$ -1
$2 \pi$ 360° 0 1 0 $\infty$ 1 $\infty$
Angle in
Angle in
Degrees
Sin Cos Tan Cot Sec Csc
0 0 1 0 $\infty$ 1 $\infty$
$\frac{\pi}{6}$ 30° $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{3}$ $\sqrt{3}$ $\frac{2\sqrt{3}}{3}$ 2
$\frac{\pi}{4}$ 45° $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ 1 1 $\sqrt{2}$ $\sqrt{2}$
$\frac{\pi}{3}$ 60° $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $\frac{\sqrt{3}}{3}$ 2 $\frac{2\sqrt{3}}{3}$
$\frac{\pi}{2}$ 90° 1 0 $\infty$ 0 $\infty$ 1
Angle in
Angle in
Degrees
Sin Cos Tan Cot
.0000 .0000 1.0000 .0000 Inf. 90° 1.5708
.0175 .0175 .9998 .0175 57.290 89° 1.5533
.0349 .0349 .9994 .0349 28.636 88° 1.5359
.0524 .0523 .9986 .0524 19.081 87° 1.5184
.0698 .0698 .9976 .0699 14.300 86° 1.5010
.0873 .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
.4863 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

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 07f9 0755
12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106
13 1139 1173 1206 1239 1271 1303 1335 1367 1399 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
Angle in
Degrees
log sin log cos log tan log cot
.0000 .... 0.000 .... .... 90° 1.5708
.0175 8.2419 9.9999 8.2419 1.7581 89° 1.5533
.0349 8.5428 9.9997 8.5431 1.4569 88° 1.5359
.0524 8.7188 9.9994 8.7194 1.2806 87° 1.5184
.0698 8.8436 9.9989 8.8446 1.1554 86° 1.5010
.0873 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
1. In formulas 14-25, $r$ denotes radius, $a$ altitude, $B$ area of base, and $s$ slant height.