PastExamLabPastExamLab

臺灣綜合大學系統 105 年度 微積分B

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第 1 題20 分極限・夾擠定理 / 微分定義
Let f(x)=x2[1−cos⁡(3x)]f(x) = x^2 \left[1 - \cos\left(\frac{3}{x}\right)\right], x≠0x \neq 0.
(1-1)5 分
Evaluate the limit. lim⁡x→0f(x)\lim\limits_{x \to 0} f(x)
(1-2)5 分
Evaluate the limit. lim⁡x→∞f(x)\lim\limits_{x \to \infty} f(x)
(1-3)10 分
Evaluate the limit. lim⁡x→0f′(x)\lim\limits_{x \to 0} f'(x)
第 2 題20 分
Let f(x)=exp⁡(1−x28)f(x) = \displaystyle\exp\left(1 - \frac{x^2}{8}\right).
(2-1)5 分
Find the intervals of increases or decreases of f(x)f(x).
(2-2)5 分
Find the intervals of concave up or concave down of f(x)f(x).
(2-3)10 分
Sketch the graph of f(x)f(x), x∈[−5,5]x \in [-5, 5]. Label the points of local maximum or minimum, points of inflection and asymptotes on the graph. (e≈2.71e \approx 2.71, e≈1.65\sqrt{e} \approx 1.65.)
第 3 題20 分
Let f(x,y)=8x3−12xy+y3f(x, y) = 8x^3 - 12xy + y^3.
(3-1)5 分
Find the tangent plane of the surface z=f(x,y)z = f(x, y) at (1,4)(1, 4).
(3-2)5 分
In what direction does f(x,y)f(x, y) have minimum rate of change at (1,4)(1, 4)?
(3-3)10 分
Find the local maximum, local minimum and saddle points of f(x,y)f(x, y).
第 4 題10 分
Evaluate the indefinite integral. ∫(ln⁡x)2x2dx\displaystyle\int \frac{(\ln x)^2}{x^2} dx
第 5 題10 分
Use third order Taylor polynomial to approximate the number 4.05\sqrt{4.05}. (Put the answer in the form 4.05≈2±1xxx±1xxxxxx±1xxxxxxxxx\sqrt{4.05} \approx 2 \pm \frac{1}{\boxed{\phantom{xxx}}} \pm \frac{1}{\boxed{\phantom{xxxxxx}}} \pm \frac{1}{\boxed{\phantom{xxxxxxxxx}}}.)
第 6 題10 分
The Cobb-Douglas production function is P=x14y12z14P = x^{\frac{1}{4}}y^{\frac{1}{2}}z^{\frac{1}{4}} subject to total budget constraint 3x+4y+5z=2403x + 4y + 5z = 240. (xx: capital, yy: labor, zz: resource.) Use Lagrange multiplier to find values of x,y,zx, y, z such that PP is maximized.
第 7 題10 分
Evaluate the double integral. ∫04∫x21x(1+y4)dydx\displaystyle\int_0^4 \int_{\sqrt{x}}^2 \frac{1}{\sqrt{x}(1 + y^4)} dy dx