∫ calculus

1. **Тодорхойлолт:** Функцийн хувь $B(t)$ дараах байдлаар өгөгдсөн: $$B(t) = \begin{cases} 4 - \frac{1}{2}t, & t < 2 \\ \sqrt{t + c}, & t \geq 2 \end{cases}$$

1. **Problem:** Find the limit $\lim_{x \to -3} h(x)$ using the graph. From the graph, at $x = -3$, $h(x) = 1$.

1. Let's start by understanding the problem: You want to know why it is important to identify when a graph is at rest. 2. In mathematics and physics, a graph is "at rest" when its

1. **State the problem:** We need to calculate the total distance travelled by the graphic from time $t=0$ to $t=15$ seconds. 2. **Understand the velocity function:** The velocity

1. **State the problem:** We are given a continuous function $f(x)$ on the interval $[0,2]$ with two integral conditions: $$\int_0^2 (f(x) + x) \, dx = 8$$

1. **Stating the problem:** Sandwich's Theorem, also known as the Squeeze Theorem, helps find the limit of a function that is "squeezed" between two other functions whose limits ar

1. **Stating the problem:** We need to evaluate the integral $$\int \sqrt{16x} \sin\left(1 + \frac{x^3}{2}\right) \, dx.$$\n\n2. **Rewrite the integral:** Note that $$\sqrt{16x} =

1. **State the problem:** We need to evaluate the integral $$\int \sqrt{16x} \sin\left(1 + x^{\frac{3}{2}}\right) \, dx.$$\n\n2. **Rewrite the integral:** Note that $$\sqrt{16x} =

1. **State the problem:** We need to find the integral $$\int \sqrt{16x} \sin\left(1 + x^{32}\right) \, dx$$. 2. **Rewrite the integral:** Note that $$\sqrt{16x} = \sqrt{16} \sqrt{

1. **Problem statement:** We have a storage tank with 10,000 litres of chemical leaking at a rate given by the derivative of the amount spilled, $$f'(t) = 400e^{-0.01t}$$ where $$t

1. **State the problem:** We want to analyze the series $$\sum_{n=1}^\infty \left( \frac{n^2}{2^n} + \frac{1}{n^2} \right)$$ and determine its behavior or sum if possible. 2. **Rec

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series with general term $a_n = \frac{n! \times 3^n}{n^n}$ using the ratio test. 2. **Recall the ratio test formula

1. **State the problem:** We want to test the convergence of the series $$\sum_{n=1}^\infty \frac{1}{n^n}$$ using the ratio test. 2. **Recall the ratio test formula:** For a series

1. **State the problem:** Find the average value of the function $f(x) = \sqrt[3]{x}$ on the interval $[1,8]$. 2. **Formula for average value of a function:** The average value $f_

1. مسئله: تابع داده شده $y = -x^3 + m x^2 - 2 m x + 3$ است. می‌خواهیم تعداد مقادیر صحیح $m$ را بیابیم که رأس سهمی در ناحیه چهارم محورهای مختصات قرار دارد. 2. ابتدا باید بدانیم رأس

1. **State the problem:** Evaluate the definite integral $$\int_{\pi/6}^{\pi/4} \left(5 - 2 \sec z \tan z\right) \, dz.$$\n\n2. **Recall the integral formulas and rules:**\n- The i

1. **State the problem:** Evaluate the integral $$\int \frac{\csc^2(x)}{\cot(x)} \, dx$$. 2. **Recall the trigonometric identities:**

1. **State the problem:** We need to find the limit of the function $f(x) = x^3 - 1$ as $x$ approaches 1. 2. **Substitute the value:** Since $f(x)$ is a polynomial, it is continuou

1. **State the problem:** We have the function $$f(x) = \frac{x^2 - 15}{x - 4}$$ and need to find where it is increasing, decreasing, and its local extrema. 2. **Find the derivativ