题目

Problem

The curve $C$ has equation

\begin{align*} y=2x^{5/2}-4x+3. \end{align*}

(a) Find $\dfrac{dy}{dx}$ , writing your answer in simplest form.

(2)

The point $P$ lies on $C$ .

Given that

(b) find the value of $k$ .

(3)

解答

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逐项微分即可。 $x^{5/2}$ 微分后，指数减 $1$ ，系数乘以原指数。

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\begin{align*} \frac{dy}{dx} =&\,2\cdot\frac52x^{3/2}-4\\ =&\,5x^{3/2}-4. \end{align*}

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梯度为 $16$ ，所以令导函数等于 $16$ ，先求出 $x$ 。再用 $x=2^k$ 比较指数。

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At $P$ ,

\begin{align*} 5x^{3/2}-4=&\,16\\ 5x^{3/2}=&\,20\\ x^{3/2}=&\,4. \end{align*}

Since the $x$ coordinate is $2^k$ ,

\begin{align*} (2^k)^{3/2}=&\,4\\ 2^{3k/2}=&\,2^2. \end{align*}

Therefore

\begin{align*} \frac{3k}{2}=&\,2\\ k=&\,\frac43. \end{align*}