115年中央大學地質碩士班-微積分詳解

 國立中央大學115學年度碩士班考試入學

系所:應用地質研究所
科目:微積分

解答:$$\textbf{a. } f(x)=3x^4+5x^2-x+7 \Rightarrow f'(x)= \bbox[red, 2pt]{12x^3+10x-1} \\ \textbf{b. }f(x)={x^5+2x^2-6 \over x^3} \Rightarrow f'(x)= {d\over dx}(x^5+2x^2-6)\cdot {1\over x^3}+ (x^5+2x^2-6) \cdot {d\over dx} \left( {1\over x^3} \right) \\\qquad ={5x^4+4x\over x^3}-{(x^5+2x^2-6)\cdot 3\over x^4} = \bbox[red, 2pt]{2x^5-2x^2+18\over x^4}$$

解答:$$f(x)= \left( {x+3\over x-2} \right)^3 \Rightarrow f'(x)=3\left( {x+3\over x-2} \right)^2\cdot {d\over dx}\left( {x+3\over x-2} \right) =3\left( {x+3\over x-2} \right)^2\cdot \left( {1\over x-2} -{x+3 \over (x-2)^2}\right) \\ \Rightarrow f'(1)=3\cdot  \left( {4\over -1} \right)^2\cdot  \left( -1-{4\over (-1)^2} \right)= 3\cdot (16)\cdot (-5)= \bbox[red, 2pt]{-240}$$

解答:$$f(x^2) = (x^3 + 5x)^3 \Rightarrow 2xf'(x^2)=3(x^3+5x)^2(3x^2+5) \Rightarrow f'(x^2)={3(x^3+5x)^2(3x^2+5) \over 2x} \\ x=\sqrt 5代入上式\Rightarrow f'(5) ={3\cdot (10\sqrt 5)^2 \cdot 20 \over 2\sqrt 5} = \bbox[red, 2pt]{3000\sqrt 5}$$

解答:$$y= 6x^5-5x^4+x^3-6x+ \cos x \Rightarrow {dy\over dx}=30x^4 -20x^3-6-\sin x \\ \Rightarrow {d^2y \over dx^2}=120x^3-60x^2-\cos x \Rightarrow {d^3y \over dx^3}= \bbox[red, 2pt]{360x^2-120x+\sin x}$$

解答:$$\textbf{a. }u=x^3 \Rightarrow du =3x^2\,dx \Rightarrow \int x^2 \sin(x^3)\,dx = \int {1\over 3}\sin u\,du =-{1\over 3}\cos u+C= \bbox[red, 2pt]{-{1\over 3}\cos(x^3)+C} \\\textbf{b. }函數  y = \sqrt{1-x^2}  表示半徑為  1 、圓心在原點的半圓上半部\\ \Rightarrow   \int_0^1 \sqrt{1-x^2} \, dx =四分之一圓面積= \bbox[red, 2pt]{{1\over 4}\pi}$$

解答:

$$\textbf{a. } y=x^2 \Rightarrow y'=2x \Rightarrow y'(1,1)=2 \Rightarrow 切線L: y=2(x-1)+1 \Rightarrow y=2x-1 \\ \quad \Rightarrow 面積A= \text{面積 } A = \int_{0}^{1} \left[ x^2 - (2x - 1) \right] dx = \int_{0}^{1} (x^2 - 2x + 1) \, dx =  \int_0^1 (x-1)^2\,dx \\ \quad = \left. \left[ {1\over 3}(x-1)^3 \right] \right|_0^1 =\bbox[red, 2pt]{1\over 3}\\ \textbf{b. }  \cases{x= cu\\y=dv} \Rightarrow  {c^2u^2 \over c^2}+{{d^2v^2\over d^2}}=1 \Rightarrow u^2+v^2=1 為一單位圓,面積為\pi \\ \quad \left|{\partial (x,y)\over \partial(u,v)} \right| =  \begin{vmatrix} c& 0\\0& d \end{vmatrix}=cd \Rightarrow {x^2\over c^2}+{y^2\over d^2}=1的面積為cd \cdot \pi= \bbox[red, 2pt]{cd\pi}$$

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解題僅供參考，碩士班歷年試題及詳解