PastExamLabPastExamLab

成功大學 107 年度 微積分B

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第 1 題10 分
Find the following limits.
lim⁡x→0sin⁡(4x3)x\lim\limits_{x \to 0} \frac{\sin(4x^3)}{x}
第 2 題10 分
Consider a circular conic tank shown below. The water is drained at the bottom at the rate of 2m32 m^3 per second. Find the rate of change of height hh when the top circular surface of water has radius 1m1m. Recall that the volume of a circular cone is 13πr2h\frac{1}{3}\pi r^2 h.
第 2 題圖表
第 3 題10 分
Evaluate the indefinite integral
∫sin⁡(x)dx\int \sin(\sqrt{x}) dx
第 4 題10 分
Find the equation of tangent plane to the surface defined by
x2+2y2+xy+ez=2x^2 + 2y^2 + xy + e^z = 2
at the point P=(1,0,0)P = (1, 0, 0).
第 5 題10 分
Evaluate the improper integral
∫1∞3x2+1x4+x2dx\int_1^{\infty} \frac{3x^2 + 1}{x^4 + x^2} dx
第 6 題10 分
Consider the function
f(x,y)={2x4+xyx2+y2;(x,y)≠(0,0)0;(x,y)=(0,0)f(x, y) = \begin{cases} \frac{2x^4 + xy}{x^2 + y^2} & ; (x,y) \neq (0, 0) \\ 0 & ; (x,y) = (0, 0) \end{cases}
Find ∂f∂x\frac{\partial f}{\partial x} at (0,0)(0, 0) if existed. If it does not exist, explain why.
第 7 題10 分
Find the point where local minimum for the function
f(x)=∫0x3−3xecos⁡(t2+1)dtf(x) = \int_0^{x^3-3x} e^{\cos(t^2+1)} dt
occurs.
第 8 題10 分
Write down the Taylor series expansion for the function
f(x)=x31+x2f(x) = \frac{x^3}{1 + x^2}
at x=0x = 0.
第 9 題10 分
Compute the integral
∫01∫x1cos⁡(y2+1)dydx\int_0^1 \int_x^1 \cos(y^2 + 1) dy dx
第 10 題10 分
Use Lagrange multiplier method to maximize the function
f(x,y,z)=2x+3y+5zf(x, y, z) = 2x + 3y + 5z
on the sphere
x2+y2+z2=19x^2 + y^2 + z^2 = 19
Note: any other method receives no credit.
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