∫ calculus

1. **Problem Statement:** Consider the infinite series $$\sum_{n=1}^{\infty} 5^{1-n} (-1)^n$$.

1. The problem is to understand and analyze the inequality $x' - 2 \leq x \leq 5$. 2. Here, $x'$ typically denotes the derivative of $x$ with respect to some variable, often time $

1. **Problem Statement:** Evaluate the integral $$\int x \sqrt{x+1} \, dx$$. 2. **Formula and Approach:** To solve integrals involving products of polynomials and roots, substituti

1. **Problem Statement:** Given $g(x) = \int_0^x f(t) \, dt$ where $f$ is a piecewise linear function with points $(0,0), (1,1), (3,4), (6,1), (9,-1), (10,1)$.

1. **Problem Statement:** Find the derivative of the function $$f(x) = \frac{1 - x}{2x}$$ using the limit definition of the derivative. 2. **Limit Definition of the Derivative:** T

1. **Problem statement:** Find the volume of the solid bounded by the surface $f(x,y)$ over the given region $R$. We will solve problems 3, 4, and 12 from Exercise 2.2.

1. **Problem statement:** Find the volume of the solid bounded by the surface $f(x,y) = 1 + 4xy$ over the region $R: 0 \leq x \leq 1, 1 \leq y \leq 3$. 2. **Formula:** The volume i

1. **Problem Statement:** Find the volume of the solid bounded by the surface for the following functions and regions: - 3. $f(x,y) = 1 + 4xy$ over $0 \leq x \leq 1$, $1 \leq y \le

1. 問題陳述： 已知函數 $y=f(x)$ 在區間 $[3,9]$ 上的圖形，求相對極小值個數 $a$、相對極大值個數 $b$、絕對極小值個數 $c$、絕對極大值個數 $d$，並計算 $a+b+c+d$。

1. 問題陳述： 我們要計算函數 $f(x) = x^2 - 9x - 3$ 在 $x \to 3$ 時的極限值 $\lim_{x \to 3} f(x)$。

1. **Stating the problem:** We need to find the right-hand limit of the function $f(x)$ as $x$ approaches $-5$ from the positive side, i.e., $\lim_{x \to -5^+} f(x)$. 2. **Understa

1. 問題陳述： 給定函數 $f(x)=\sqrt[3]{x^3} - \sqrt[3]{x}$，求介於 $0$ 與 $\sqrt[3]{2^3}=2$ 之間的一實數 $c$，使得

1. Problem 8: Evaluate $(f \circ g)'(1)$ given $f(1)=2$, $g(1)=2$, $f'(1)=3$, $g'(1)=2$, $f'(2)=1$, and $g'(2)=3$. The chain rule states:

1. The Mean Value Theorem (MVT) states that for a function $f$ continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, there exists at least one

1. **Problem 7:** Find the derivative of $f(x) = x^2 + 7x - 2$ and evaluate $f'(0)$. 2. **Formula:** The derivative of $x^n$ is $nx^{n-1}$ and the derivative of $ax$ is $a$.

1. **Problem 4:** Find the equation of the tangent line to the curve $h(t) = t^3 - 9t - 1$ at the point $(3, -1)$. 2. **Step 1:** Find the derivative $h'(t)$ which gives the slope

1. **Problem 4:** Find the equation of the tangent line to the curve $h(t) = t^3 - 9t - 1$ at the point $(3, -1)$. 2. The formula for the tangent line at a point $t=a$ is:

1. The problem asks us to sketch the derivative of a graph with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. 2. The original graph behavior is:

1. Το πρόβλημα ζητά να υπολογίσουμε το ολοκλήρωμα της συνάρτησης \( \sin(x) \) στο διάστημα από 0 έως \( \pi \).\n\n2. Η βασική φόρμουλα για το ορισμένο ολοκλήρωμα είναι:\n$$\int_a

1. **State the problem:** Find the equation of the tangent line to the graph of $y = x \sin x$ at $x = \frac{\pi}{2}$. The point on the graph is $\left(1, -1\right)$, but the tange

1. **State the problem:** We have a piecewise function $$f(x) = \begin{cases} 2x + a, & x \leq 3 \\ bx + 4, & x > 3 \end{cases}$$