- Suppose f(x)=Ax5+Bx4+Cx+101≤101 for all x and f(1)=1. Determine the values of A,B and C.
- Suppose f\left(x\right) is a differentiable function defined on \left(-\infty,\infty\right), satisfying f\left(x+y\right)=f\left(x\right)+f\left(y\right), for every x and y. Show that f''\left(x\right)=0.
- Evaluate the definite integral \displaystyle\int_0^{\pi}x\sin xdx.
- Estimate the value of \ln1.1 so that the error is smaller than 10^{-5}.
- Evaluate the volume of the solid bounded by the surface
2x^2+3y^2+3z^2=6
- Suppose F\left(x,y\right) and G\left(x,y\right) are two differentiable functions defined on \mathbb{R}^2=\left\{\left(x,y\right)|x,y\in\left(-\infty,\infty\right)\right\} so that the gradients \displaystyle\nabla F\left(x,y\right)=\left(\frac{\partial F}{\partial x}\left(x,y\right),\frac{\partial F}{\partial y}\left(x,y\right)\right), \displaystyle\nabla G\left(x,y\right)=\left(\frac{\partial G}{\partial x}\left(x,y\right),\frac{\partial G}{\partial y}\left(x,y\right)\right) are always parallel in the sense that, for every \left(x,y\right), there is a number \lambda, possibly dependent of the point \left(x,y\right), satisfying
\nabla F\left(x,y\right)=\lambda\cdot\nabla G\left(x,y\right).
Is it true that F must be a constant multiple of G? Prove it or give a counter example. - Suppose f\left(x\right) is continuous function defined on \left[-1,1\right] so that \displaystyle\int_a^bf\left(x\right)dx\geq0 for every a,b\in\left[-1,1\right], a\leq b. Give a reason why we can or can not conclude that f\left(x\right)\geq0, for every x\in\left[-1,1\right].
- Solve the differential equation
y'=100y-y^2, y\left(0\right)=1.
- Evaluate the integral \displaystyle\iint_D\left(x^2-y^2\right)dxdy, where
D=\left\{\left(x,y\right)|0\leq x+y\leq8,~0\leq x-y\leq4\right\}.
- Determine the maximum of the function
f\left(x,y,z\right)=3x^2+2y^2+z^2
defined on the surface\left\{\left(x,y,z\right)|2x^2+27y^2+10z^2=12\right\}.
訣竅
逐步分析各參數之值。解法
假若 A>0,那麼 limx→∞f(x)=∞;假若 A<0,那麼 limx→−∞f(x)=∞,故 A 僅能等於 0。
再者,無法 B=C=0,否則 f(1)≠1。假若 B=0 但 C≠0,則 f 為一次式,故不可能總是不超過 101,因此 B≠0。進一步的,假若 B>0,那麼有 limx→±∞f(x)=∞,不合於條件。因此 B<0。
此外,我們也有 f(1)=1,因此 B+C=−100,從而現在有
f(x)=Bx4−(B+100)x+101
為了確保 f 總是不超過 101,我們考慮 f 的最大值,故考慮方程式 f′(x)=4Bx3−(B+100)=0,如此可解得x=3√B+1004B
又因 f″ (因 B<0),故在該位置確實達到最大值,檢驗可知\displaystyle f\left(\sqrt[3]{\frac{B+100}{4B}}\right)=\sqrt[3]{\frac{B+100}{4B}}\left(\frac{B+100}{4B}-B-100\right)+101\leq101
等價於\displaystyle\sqrt[3]{B+100}\left(\frac{25}B-B-\frac{399}4\right)\geq0
也等價於 \left(B+100\right)\left(B+100\right)\left(4B-1\right)\geq0,從而利用 B<0 解得 B=-100,而 C=0。訣竅
此為經典的柯西函數方程,將此方程解出後可直接求二階導函數。解法
取 x=y=0 時有 f\left(0\right)=2f\left(0\right),故 f\left(0\right)=0。再者,取 y=-x 則有 0=f\left(0\right)=f\left(x\right)+f\left(-x\right),從而 f\left(-x\right)=-f\left(x\right),此表明 f 為奇函數。
我們先使用數學歸納法證明 f\left(n\right)=nf\left(1\right),此處 n\in\mathbb{N}。明顯當 n=1 時成立,現設 n=k 時有 f\left(k\right)=kf\left(1\right),那麼有
\displaystyle f\left(k+1\right)=f\left(k\right)+f\left(1\right)=kf\left(1\right)+f\left(1\right)=\left(k+1\right)f\left(1\right)
至此可確知 f 在正整數的取值。下一步,我們考慮 f 在正有理數上的取值。我們首先同樣由數學歸納法注意到下列的等式
\displaystyle f\left(\sum_{k=1}^nx_k\right)=\sum_{k=1}^nf\left(x_k\right)
首先取 \displaystyle x_1=\cdots=x_n=\frac1n,那麼可以得到 \displaystyle f\left(\frac1n\right)=\frac1nf\left(1\right),從而有 \displaystyle f\left(\frac{m}n\right)=\frac{m}nf\left(1\right)。再透過奇函數的的特性,我們確認了:對於任何有理數 q\in\mathbb{Q} 恆有 f\left(q\right)=qf\left(1\right)。最後,由於 f 在 \mathbb{R} 上可導,故 f 必然在 \mathbb{R} 上連續。對於任何實數 x 我們可考慮一有理數數列 \left\{q_n\right\}_{n=1}^{\infty} 趨近於 x,那麼由 f 的連續性,我們有
\displaystyle f\left(x\right)=f\left(\lim_{n\to\infty}q_n\right)=\lim_{n\to\infty}f\left(q_n\right)=\lim_{n\to\infty}q_nf\left(1\right)=xf\left(1\right)
這表明 f 為通過原點的一次函數,故其二階導函數恆為零,證明完畢。訣竅
使用分部積分法即可。解法
使用分部積分法可知\displaystyle\int_0^\pi x\sin xdx=-\int_0^\pi xd\cos x=-x\cos x\Big|_0^{\pi}+\int_0^{\pi}\cos xdx=\pi+\sin x\Big|_0^{\pi}=\pi
訣竅
由基本的無窮等比級數出發來思考,隨後使用積分導出自然對數函數的展開式並用以計算近似值,最後使用交錯級數的誤差估計獲得所需的近似值。解法
由無窮等比級數可知\displaystyle\frac1{1+x}=\sum_{k=0}^{\infty}\left(-1\right)^kx^k
同取積分可知\displaystyle\ln\left(1+x\right)=\int_0^x\sum_{k=0}^{\infty}\left(-1\right)^kt^kdt=\sum_{k=0}^{\infty}\left(-1\right)^k\int_0^xt^kdt=\sum_{k=0}^{\infty}\frac{\left(-1\right)^k}{k+1}x^{k+1}
取 x=0.1 可得\displaystyle\ln1.1=\sum_{k=0}^{\infty}\frac{\left(-1\right)^k}{\left(k+1\right)10^k}
當 k=5 時有 \displaystyle\frac1{6\cdot10^5}<10^{-5},故所求的近似值為\displaystyle\ln1.1\approx\sum_{k=0}^4\frac{\left(-1\right)^k}{\left(k+1\right)10^k}=1-\frac1{20}+\frac1{300}-\frac1{4000}+\frac1{50000}=\frac{285931}{300000}=0.95310\overline{3}
【註】 使用電子計算器可知
\ln1.1\approx0.0953102
訣竅
使用橢球體積公式即可;亦可運用旋轉體表面積公式;也能直接計算三重積分求解。解法一
方程可寫為\displaystyle\frac{x^3}3+\frac{y^2}2+\frac{z^2}2=1
可辨識出此為橢球方程,故所求體積為 \displaystyle\frac43\pi\sqrt3\cdot\sqrt2\cdot\sqrt2=\frac{8\sqrt3\pi}3。解法二
當 z=0 時橢球方程被 xy 平面截出一橢圓 2x^2+3y^2=6,可觀察到該橢球為繞 x 軸所形成的。為了應用旋轉體體積公式,我們考慮函數 f\left(x\right)=\sqrt{\left(6-2x^2\right)/3},其中範圍為 x\in\left[-\sqrt3,\sqrt3\right],如此可知\displaystyle V=\int_{-\sqrt3}^{\sqrt3}\pi\cdot\sqrt{\frac{6-2x^2}3}^2dx=\frac{2\pi}3\int_{-\sqrt3}^{\sqrt3}\left(3-x^2\right)dx=\left.\frac{2\pi}3\left(3x-\frac{x^3}3\right)\right|_{-\sqrt3}^{\sqrt3}=\frac{8\sqrt3\pi}3
解法三
考慮集合\Omega=\left\{\left(x,y,z\right)\in\mathbb{R}^3:~2x^2+3y^2+3z^2\leq6\right\}
那麼所求的體積可表達如下\displaystyle\begin{aligned}V&=\iiint_{\Omega}dV=\int_{-\sqrt3}^{\sqrt3}\int_{-\sqrt{\left(6-2x^2\right)/3}}^{\sqrt{\left(6-2x^2\right)/3}}\int_{-\sqrt{\left(6-2x^2-3y^2\right)/3}}^{\sqrt{\left(6-2x^2-3y^2\right)/3}}dzdydx\\&=\frac{8\sqrt3}3\int_0^{\sqrt3}\int_0^{\sqrt{\left(6-2x^2\right)/3}}\sqrt{6-2x^2-3y^2}dydx\end{aligned}
運用極座標變換,令 \left\{\begin{aligned}&x=\sqrt3r\cos\theta\\&y=\sqrt2r\sin\theta\end{aligned}\right.,其中變數範圍為 \left\{\begin{aligned}&0\leq r\leq1\\&0\leq\theta\leq\frac\pi2\end{aligned}\right.,那麼所求可改寫並計算如下\displaystyle V=\frac{8\sqrt3}3\int_0^{\frac\pi2}\int_0^1\sqrt{6-6r^2}\cdot\sqrt6rdrd\theta=16\sqrt3\cdot\frac\pi2\int_0^1r\sqrt{1-r^2}dr=8\sqrt3\pi\cdot\left.-\frac13\left(1-r^2\right)^{\frac32}\right|_0^1=\frac{8\sqrt3\pi}3
訣竅
藉由簡單的試誤與推敲即可確認本命題錯誤。解法
簡單考慮函數 F\left(x,y\right)=x+y 與 G\left(x,y\right)=x+y+1,那麼取 \lambda=1 可知\nabla F\left(x,y\right)=\left(1,1\right)=1\cdot\left(1,1\right)=\lambda\cdot\nabla G\left(x,y\right)
然而 F 並非 G 的常數倍。訣竅
透過連續函數的特性來使用反證法處理。解法
假設存在一點 x_0\in\left[-1,1\right] 使得 f\left(x_0\right)<0,那麼由 f 的連續性可知存在一正數 \delta 使得\displaystyle x\in\left(x_0-\delta,x_0+\delta\right)\cap\left[-1,1\right]~~\Longrightarrow~~\left|f\left(x\right)-f\left(x_0\right)\right|<-\frac{f\left(x_0\right)}2
故取 \displaystyle\left[a,b\right]=\left[x_0-\frac{\delta}2,x_0+\frac{\delta}2\right]\cap\left[-1,1\right],那麼便有\displaystyle\int_a^bf\left(x\right)dx\leq\int_a^b\frac{f\left(x_0\right)}2dx=\frac{\left(b-a\right)f\left(x_0\right)}2<0
矛盾。故每一點的函數值皆非負。訣竅
運用分離變量法處理;亦可將之視為白努利方程來使用變數代換法處理。解法一
移項後有\displaystyle\left(\frac1{100-y}+\frac1{y}\right)dy=\frac{100dy}{100y-y^2}=100dt
在 \left[0,t\right] 上同取定積分可知\displaystyle\ln\frac{y}{100-y}=100t-\ln99
兩邊取自然指數可得\displaystyle\frac{y}{100-y}=\frac1{99}e^{100t}
從而解得\displaystyle y\left(t\right)=\frac{100e^{100t}}{99+e^{100t}}=\frac{100}{1+99e^{-100t}}
解法二
令 \displaystyle z\left(t\right)=\left[y\left(t\right)\right]^{1-2}=\frac1{y\left(t\right)},那麼便有\displaystyle\frac{dz}{dt}=-\frac1{\left(y\left(t\right)\right)^2}\frac{dy}{t}=-\frac{100}{y\left(t\right)}+1=-100z\left(t\right)+1
故兩邊同乘以 e^{100t} 可得\displaystyle\frac{d}{dt}\left(e^{100t}z\left(t\right)\right)=e^{100t}\frac{dz}{dt}+100e^{100t}z\left(t\right)=e^{100t}
隨後取積分便有\displaystyle e^{100t}z\left(t\right)-1=\frac{e^{100t}-1}{100}
因此\displaystyle z\left(t\right)=\frac{1+99e^{-100t}}{100}
如此所求的函數為\displaystyle y\left(t\right)=\frac{100}{1+99e^{-100t}}
訣竅
運用變數變換求解即可。解法
考慮變數變換 \left\{\begin{aligned}&u=x+y\\&v=x-y\end{aligned}\right.,那麼積分範圍化為 \left\{\begin{aligned}&0\leq u\leq8\\&0\leq v\leq4\end{aligned}\right.。再者,其 Jacobian 行列式計算如下\displaystyle\frac{\partial\left(x,y\right)}{\partial\left(u,v\right)}=\left|\begin{matrix}\displaystyle\frac{\partial x}{\partial u}&\displaystyle\frac{\partial x}{\partial v}\\\displaystyle\frac{\partial y}{\partial u}&\displaystyle\frac{\partial y}{\partial v}\end{matrix}\right|=\left|\begin{matrix}\displaystyle\frac{\partial u}{\partial x}&\displaystyle\frac{\partial u}{\partial y}\\\displaystyle\frac{\partial v}{\partial x}&\displaystyle\frac{\partial v}{\partial y}\end{matrix}\right|^{-1}=\left|\begin{matrix}1&1\\1&-1\end{matrix}\right|^{-1}=-\frac12
如此所求的重積分可改寫並計算如下\displaystyle\iint_D\left(x^2-y^2\right)dxdy=\int_0^4\int_0^8uv\cdot\left|-\frac12\right|dudv=\frac12\left(\int_0^4vdv\right)\left(\int_0^8udu\right)=\frac12\cdot8\cdot32=128
訣竅
運用初等不等式求解即可。解法
由限制條件可以注意到\displaystyle3x^2=18-\frac{81y^2}2-15z^2
如此給定的函數 f 可透過這樣的限制條件改寫如下\displaystyle f\left(x,y,z\right)=3x^2+2y^2+z^2=18-\frac{81y^2}2-15z^2+2y^2+z^2=18-\frac{77y^2}2-14z^2\leq18
此時等號成立條件為 y=z=0 而 x=\pm\sqrt6。
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