Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/25

 

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$\scriptstyle {=\pi r^{2}}$.

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

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