114年 - 114 國立嘉義大學_轉學生招生考試試題_理工學院：微積分#143766

一、 Evaluate  \( \lim_{x\to 0} \frac{\sqrt{x^2+100}-10}{x^2} \) .

二、Find \(\lim_{t\to 0} \frac{\tan t \sec 2t}{3t}\) .

三、 Find the horizontal asymptotes of the graph of

\(f(x) = \frac{x^3-2}{|x|^3+1}\) .

四、 Find the equation of the tangent line to the curve =√ x at  x= 4.

五、Find \( \frac{d^2y}{dx^2} \) if \( 2x^3 - 3y^2 = 8 \)..

六、 Let f(x)= [x], the greatest integer less than or equal to x. ) Evaluate \( \int_{1}^{4} f(x)dx. \)

七、 Compute the arc length of the curve \( y = \frac{x^{3/2}}{3} \) from x = 0 to x = 4.

八、 Determine whether the improper integral \( \int_{1}^{\infty} \frac{1}{x^2} dx \) converges. If so, find its value.

九、 Evaluate the indefinite integral \( \int (x^2 + 2x)e^x dx \) .

十、 Find the area enclosed between the two curves: y=8−x² and y=x² .