∫ calculus

1. The problem asks to find the value of a line integral over a curve $C$. However, the exact integral expression and the curve $C$ are not provided in the question. 2. Generally,

1. The problem is to find the derivative of a function, which means determining the rate at which the function's value changes with respect to its input variable. 2. The formula fo

1. Let's start by stating the problem: We want to understand what a differentiable function is and how to determine if a function is differentiable. 2. A function $f(x)$ is said to

1. **State the problem:** We need to find the absolute maximum value of the function $f(x) = x e^{-x/2}$ on the interval $[0,6]$. 2. **Recall the method:** To find absolute extrema

1. Let's start by stating the problem: We want to understand what a continuous differentiable function is. 2. A function $f(x)$ is called \textbf{continuous} at a point $x=a$ if th

1. **State the problem:** We want to use Newton's Method to approximate the zero(s) of the function $$f(x) = x - 2\sqrt{x} + 1$$ until two successive approximations differ by less

1. **State the problem:** Find the derivative of the function $$f(x) = 2^x \log_3 \left(7^{x^2 - 4}\right)$$. 2. **Rewrite the function:** Use the logarithm power rule $$\log_b (a^

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = x^2 \cos\left(\sqrt{x^3 - 1} + 2\right)$$

1. **State the problem:** Find the equation of the tangent line to the curve defined implicitly by $$x^3 - x \ln(y) + y^3 = 2x + 5$$ at the point $(2,1)$. 2. **Recall the formula:*

1. **Problem Statement:** Find the area under the curve of the function $f(x) = 11x^3 - 8x^2 + 5x + 7$ on the interval $\left[-\frac{3}{4}, \frac{9}{4}\right]$ using Left Hand Rule

1. **State the problem:** We need to find real numbers $a$ and $b$ such that the piecewise function $$

1. **State the problem:** We need to evaluate the limit $$\lim_{x \to 1} f(x)$$ where $$f(x) = \begin{cases} \frac{\sqrt{x} - 1}{x - 1} & x > 1 \\ 8 & x = 1 \\ \frac{2x - 2}{x^2 +

1. **Problem Statement:** Given that $\int f(x)\,dx = g(x) + c_1$ and $\int g(x)\,dx = \ln\left(\frac{1}{x}\right) + c_2$, find the value of $f(2) + g(2)$.\n\n2. **Understanding th

1. Sketch and analyze $f(x) = \frac{1}{x}$. - Problem: Graph $f(x) = \frac{1}{x}$ and identify discontinuities.

1. **Тодорхой интегралын асуудал 22:**\n\nӨгөгдсөн: $g(x) = \int_a^x f(t) dt$, $f$ функцийн графикаар өгөгдсөн.\n\n(а) $g(x)$ функцийн максимум, минимум утгуудыг олох:\n\n- $g'(x)

1. **Problem Statement:** We analyze the behavior of the function $f(x)$ at points $A$, $B$, $C$, $D$, and $E$ on its graph, focusing on the sign changes of its first and second de

1. The problem involves understanding the behavior of the first and second derivatives of a function and what these changes in sign indicate about the function's graph. 2. Recall t

1. **Problem Statement:** We analyze the behavior of the function $f(x)$ at critical points $A$, $B$, $C$, $D$, and $E$ based on the graph description. 2. **Key Concepts:**

1. **State the problem:** Evaluate the integral $$\int \frac{5x^4}{x^5 + 7} \, dx$$ using substitution. 2. **Choose substitution:** Let $$u = x^5 + 7$$.

1. We are asked to evaluate the integral $$\int \frac{5x^4}{x^5 + 7} \, dx$$ and find a substitution to rewrite the integrand as $$\frac{1}{u} \, du$$. 2. Let us choose the substit

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^3}{e^3 - 4x^5} \, dx$$ and remember to include the constant of integration. 2. **Identify a substitution:*