所屬科目：研究所、轉學考(插大)-微積分
1. (60%) Answer the following questions.(a) Find \( \lim_{x\to 0}\frac{e^{29x} - 1}{\sin(31x)}. \)
(b) Find \( \lim_{x\to -\infty}\frac{41x^4 - 43x + 47}{53x^4 + 59x^2 - 61}. \)
(c) Find \( \lim_{x\to 0}\frac{e^{3x} - 1 - 3x}{x^2} \).
(d) Find the domain of \( f(x) = \sqrt{7 - x} + \ln x \).
(e) Given \( y = x^{11} + \ln x \), find \( \frac{dx}{dy} \).
(f) Given \( y = xe^{x^3} \), find \( \frac{dy}{dx} \).
(g) Given \( z = 7y^3 \) and \( y = 6x^2 + 8x \), find \( \frac{dz}{dx} \).
(h) Given \( y = \frac{x^4 + 7e^x}{x^2 + 1} \), find \( \frac{dy}{dx} \).
(i) Given \( z = x^{2}y + 3xy^{2} + 5y \), find the total derivative \( dz \).
(j) Evaluate \( \int_{-3}^{6}(|x - 4|)dx \)
(k) Evaluate \( \int_0^\infty 101xe^{-3x}dx \)
(1) Evaluate \( \int{1}^{6}\int{0}^{3}(3x^{2}y + 7xy^{2} + 9x)dxdy. \)
(a) Find \( \lim_{x\to \infty}(\sqrt{x^2 + 6x} - x) \).
(b) Find the critical points of the function \( f(x) = x^3 - 36x^2 + 429x - 37 \).
(c) Find the absolute maximum and minimum points of \( f(x, y) = x^2 + 13y - 12xy \) over the region bounded by \( y = x, y = 0 \), and \( x = 10 \).
(d) Approximate \( f(x, y) = \sqrt{x + y} \) around \( (x, y) = (1, 0) \) using a 2-degree Taylor polynomial.
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