PastExamLabPastExamLab

成功大學 104 年度 微積分A

PDF
第 1 題10 分
Find the following limits:
(a)5 分
lim⁡x→π4sec⁡3x−2tan⁡x1+cos⁡4x\lim\limits_{x \to \frac{\pi}{4}} \frac{\sec^3 x - 2\tan x}{1 + \cos 4x}
(b)5 分
lim⁡x→∞(x+1x+3)x−2\lim\limits_{x \to \infty} \left(\frac{x+1}{x+3}\right)^{x-2}
第 2 題10 分
Define f(x)={1x[(1+x)n−1]if x≠0Aif x=0f(x) = \begin{cases} \frac{1}{x}[(1+x)^n - 1] & \text{if } x \neq 0 \\ A & \text{if } x = 0 \end{cases}
(a)5 分
Find value of A such that f is continuous at x=0x = 0
(b)5 分
Is f(x)f(x) differentiable at x=0x = 0? Explain.
第 3 題12 分
Evaluate the following integrals:
(a)6 分
∫x+1(x−1x+1)dx\int \sqrt{x+1} \left(x - \frac{1}{\sqrt{x}} + 1\right) dx
(b)6 分
∫1∞ln⁡xx4dx\int_1^{\infty} \frac{\ln x}{x^4} dx
第 4 題10 分
Find dudt\frac{du}{dt} at t=2t = \sqrt{2} if u=ln⁡(sin⁡(xy−12))u = \ln\left(\sin(xy-\frac{1}{2})\right), when x=t22x = \frac{t^2}{2}, y=t2−1y = \sqrt{t^2-1}.
第 5 題10 分
Find the area of the polar region R common to the two regions bounded by the curve r=−6cos⁡θr = -6\cos\theta and r=2−2cos⁡θr = 2 - 2\cos\theta
第 6 題10 分
Find the volume of the solid that lies under the cone z=x2+y2z = \sqrt{x^2 + y^2}, about the xy-plane, and inside the cylinder x2+y2=2xx^2 + y^2 = 2x.
第 7 題10 分
If f(x)=ln⁡(x+12x+1)f(x) = \ln\left(\frac{x+1}{2x+1}\right), find f(n)(0)f^{(n)}(0).
第 8 題10 分
For what value(s) of aa does ∫1∞axx2+1−12xdx\int_1^{\infty} \frac{ax}{x^2+1} - \frac{1}{2x} dx converge?
第 9 題10 分
Determine and classify the stationary points of the function f(x,y)=x2+y3+6xy−7x−6yf(x,y) = x^2 + y^3 + 6xy - 7x - 6y.
第 10 題8 分
Convert ∫02π∫02∫r4−r23rdzdrdθ\int_0^{2\pi} \int_0^{\sqrt{2}} \int_r^{\sqrt{4-r^2}} 3rdzdrd\theta to
(a)4 分
rectangular coordinates with order of integration dzdxdydzdxdy,
(b)4 分
spherical coordinates with order of integration dρdϕdθd\rho d\phi d\theta. (Note: do not evaluate the integrals both in (a) and (b))
廣告區域 (Google AdSense)