Cálculo preguntas y respuestas

(a) Prove that the set of natural numbers is countable. (b) Find the limit point/points of: \( \mathrm{A}=\left\{1,-1,1 \frac{1}{2},-1 \frac{1}{2}, 1 \frac{1}{3},-1 \frac{1}{3}, \ldots\right\} \)

Bài 2. Tính tích phân \( I=\iint_{D} x^{2} y d x d y \), trong đó D là miền đóng và bị chặn giới hạn bởi tam giác OAB, biết \( O(0 ; 0), A(1 ; 0), B(2 ; 2) \).

BÀI TẬP Bài 1. Xác định cận lấy tích phân trong hệ tọa độ cực của tích phân: \( I=\iint_{D} f(x ; y) d x d y \) với miền \( D=\left\{(x ; y) \mid x^{2}+y^{2}-2 x \leq 0 ; y \leq 0\right\} \)

\( f(x)=3 x^{2}+6 x-45 \) A. The parabola opens downward and has a maximum value of -48 . B. The parabola opens upward and has a minimum value of (c) What is the range of \( f(x) \) ? The range of \( f(x) \) is \( [-48, \infty) \). (Type your answer in interval notation. Use integers or fractions for any nungibers in the expression.) (d) On what interval is the function increasing? The function is increasing on \( (-1, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) On what interval is the function decreasing? The function is decreasing on \( \square \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

\( \left. \begin{array} { l } { \lim _ { h \rightarrow 0 } \frac { f ( a + h ) - f ( a ) } { h } } \\ { f ( x ) = x ^ { 2 } - 1 , \quad a = 0 } \end{array} \right. \)

8. \( \lim _{x \rightarrow 0} \frac{e^{3+x}-e^{3}}{x} \) 10. \( \lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x} \)

یک تابع زیر را با استفاده از تقسیم طویل انتگرال بگیرید: \( \frac{x^{3} + 2x^{2} + 1}{x + 1} \)

Hallar \( y_{1} \) \( y=\frac{x^{2}}{\sqrt[3]{x^{2}+4}} \)

1. \( \lim _{x=\infty}\left(2 \frac{1}{x}\right)\left(3+\frac{1}{x}\right)= \)

If \( g(x) = e^{2x} \), find \( g(3) \).