Page:Calculus Made Easy.pdf/277

Exercises VI. (p. 73.)

(1) ${\dfrac {x}{\sqrt {x^{2}+1}}}$.

(2) ${\dfrac {x}{\sqrt {x^{2}+a^{2}}}}$.

(3) $-{\dfrac {1}{2{\sqrt {(a+x)^{3}}}}}$.

(4) ${\dfrac {ax}{\sqrt {(a-x^{2})^{3}}}}$.

(5) ${\dfrac {2a^{2}-x^{2}}{x^{3}{\sqrt {x^{2}-a^{2}}}}}$.

(6) ${\dfrac {{\frac {3}{2}}x^{2}\left[{\frac {8}{9}}x\left(x^{3}+a\right)-\left(x^{4}+a\right)\right]}{(x^{4}+a)^{\frac {2}{3}}(x^{3}+a)^{\frac {3}{2}}}}$.

(7) ${\dfrac {2a\left(x-a\right)}{(x+a)^{3}}}$.

(8) ${\frac {5}{2}}y^{3}$.

(9) ${\dfrac {1}{(1-\theta ){\sqrt {1-\theta ^{2}}}}}$.

 

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Exercises VII. (p. 75.)

(1) ${\dfrac {dw}{dx}}={\dfrac {3x^{2}\left(3+3x^{3}\right)}{27\left({\frac {1}{2}}x^{3}+{\frac {1}{4}}x^{6}\right)^{3}}}$.

(2) ${\dfrac {dv}{dx}}=-{\dfrac {12x}{{\sqrt {1+{\sqrt {2}}+3x^{2}}}\left({\sqrt {3}}+4{\sqrt {1+{\sqrt {2}}+3x^{2}}}\right)^{2}}}$.

(3) ${\dfrac {du}{dx}}=-{\dfrac {x^{2}\left({\sqrt {3}}+x^{3}\right)}{\sqrt {\left[1+\left(1+{\dfrac {x^{3}}{\sqrt {3}}}\right)^{2}\right]^{3}}}}$.

 

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Exercises VIII. (p. 91.)

(2) $1.44$.

(4) ${\dfrac {dy}{dx}}=3x^{2}+3$; and the numerical values are: $3$, $3{\frac {3}{4}}$, $6$, and $15$.

(5) $\pm {\sqrt {2}}$.

(6) ${\dfrac {dy}{dx}}=-{\dfrac {4}{9}}{\dfrac {x}{y}}$. Slope is zero where $x=0$; and is $\mp {\dfrac {1}{3{\sqrt {2}}}}$ where $x=1$.

(7) $m=4$, $n=-3$.

(8) Intersections at $x=1$, $x=-3$. Angles $153^{\circ }\;26'$, $2^{\circ }\;28'$.

(9) Intersections at $x=3.57$, $x=3.50$. Angles $16^{\circ }\;16'$.

(10) $x={\tfrac {1}{3}}$, $y=2{\tfrac {1}{3}}$, $b=-{\tfrac {5}{3}}$.

 

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