∫ calculus

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 statement:** Differentiate the following functions with respect to $x$. 2. **Recall differentiation rules:**

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{3x^2}{x^4 - 1}$$. 2. **Recall the rule for limits at infinity:** When evaluating limits of rational functio

1. **State the problem:** We are given that $$\lim_{x \to 2} 3f(x) = 10$$ and need to find $$\lim_{x \to 2} \left(f(x) + 2x^2\right)$$. 2. **Use the limit properties:** The limit o

1. **State the problem:** Evaluate the integral $$\int \left(x^6 - \frac{1}{x^5} + \frac{3}{3x}\right) \, dx$$ and check the answer by differentiating. 2. **Rewrite the integral:**

1. **State the problem:** We need to find the derivative of the function $$y = 5x^6 - \sec^{-1}(x)$$. 2. **Recall the derivative rules:**

1. **State the problem:** Find the second derivative $y''$ for the implicit differential equation $$\frac{5}{x} - \frac{3}{y} = 8.$$\n\n2. **Rewrite the equation:** $$\frac{5}{x} -

1. **State the problem:** Find the second derivative $y''$ for the differential equation $5x - 3y = 8$. 2. **Rewrite the equation:** The equation is $5x - 3y = 8$.

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ if $y = \ln\left(\frac{x^3}{x+1}\right)$.\n\n2. **Recall the formula:** The derivative of $\ln(u)$ with res

1. **State the problem:** Find the derivative $y'$ of the function $$y = 7x - \ln(5x)$$. 2. **Recall the derivative rules:**

1. **State the problem:** Find the derivative $y'$ of the function $$y = \frac{\ln(3x)}{7x}.$$\n\n2. **Recall the formula:** To differentiate a quotient $\frac{u}{v}$, use the quot

1. **State the problem:** Find the derivative $y'$ of the function $$y = \ln\left((x^4 + 8)^5\right).$$ 2. **Recall the formula:** The derivative of $\ln(u)$ with respect to $x$ is

1. **State the problem:** Find the derivative $y'$ of the function $y = (\ln(x))^6$. 2. **Recall the formula:** To differentiate a composite function like $y = [u(x)]^n$, use the c

1. Problem: Find the limit $\lim_{x \to -5} (3x - 7)$. Since this is a polynomial, the limit is the value of the function at $x = -5$.

1. **State the problem:** We need to find the derivative of the function $f(x) = \sqrt{x-3}$ using the definition of the derivative. 2. **Recall the definition of the derivative:**

1. The problem asks to find and graph the orthogonal trajectories of the family of curves given by $$x^2 - y^2 = c_1$$. 2. The given family of curves satisfies the implicit equatio

1. **Problem Statement:** We need to find the radius $r$ and cylindrical height $h$ of a reservoir shaped as a right circular cylinder with a hemispherical top that maximizes the v

1. **Problem:** Given the implicit equation $$x^2 + 2xy + y^2 = 1$$, find $$\frac{dy}{dx}$$. 2. **Problem:** Given the implicit equation $$x + xy + y = 1$$, find the second derivat

1. The problem is to find the indefinite integral $$\int \frac{3x - 4}{2x - 4} \, dx$$. 2. To solve this, we use the method of algebraic manipulation and substitution. First, simpl

1. **State the problem:** We need to find the indefinite integral of the function $1 + \frac{u}{2} + u^2$ with respect to $u$. 2. **Recall the integral rules:**

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,