Page:Calculus Made Easy.pdf/210

been told at what height above the origin the curve will begin, when $x=0$. So we write, as the equation of the curve drawn in Fig. 51,

$y={\tfrac {1}{10}}x^{2}+C$.

Fig. 51.

 

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Exercises XVI. (See page 262 for Answers.)

(1) Find the ultimate sum of ${\frac {2}{3}}+{\frac {1}{3}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{24}}+$ etc.

(2) Show that the series $1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+{\frac {1}{7}}$ etc., is convergent, and find its sum to $8$ terms.

(3) If $\log _{\epsilon }(1+x)=x-{\dfrac {x^{2}}{2}}+{\dfrac {x^{3}}{3}}-{\dfrac {x^{4}}{4}}+$ etc, find $\log _{\epsilon }1.3$.

(4) Following a reasoning similar to that explained in this chapter, find $y$,

$(a){\text{if}}\;{\frac {dy}{dx}}={\tfrac {1}{4}}x;(b){\text{if}}\;{\frac {dy}{dx}}=\cos x.$

(5) If ${\dfrac {dy}{dx}}=2x+3$, find $y$.