题目

Problem

A curve has equation $y=f(x)$ .

Given that

\begin{align*} f'(x)=27x^2-\frac{21x^3-5x}{2\sqrt{x}},\qquad x>0, \end{align*}

and that the curve passes through the point $(9,10)$ ,

find $f(x)$ , giving each term in its simplest form.

(6)

解答

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先把含根号的分式拆开，并写成指数形式。这里要特别小心负号：整个分式前面有一个减号。积分后用点 $(9,10)$ 求常数。

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Since $\sqrt{x}=x^{1/2}$ ,

\begin{align*} \frac{21x^3-5x}{2\sqrt{x}} =&\,\frac{21x^3}{2x^{1/2}}-\frac{5x}{2x^{1/2}}\\ =&\,\frac{21}{2}x^{5/2}-\frac52x^{1/2}. \end{align*}

\begin{align*} f'(x) =&\,27x^2-\frac{21}{2}x^{5/2}+\frac52x^{1/2}. \end{align*}

Integrating,

\begin{align*} f(x) =&\,9x^3-3x^{7/2}+\frac53x^{3/2}+c. \end{align*}

Use $f(9)=10$ :

\begin{align*} 10 =&\,9(9)^3-3(9)^{7/2}+\frac53(9)^{3/2}+c\\ =&\,6561-6561+45+c\\ =&\,45+c. \end{align*}

Hence

\begin{align*} c=-35. \end{align*}

Therefore

\begin{align*} f(x)=9x^3-3x^{7/2}+\frac53x^{3/2}-35. \end{align*}