∫ calculus

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ for the functions using the definition of derivative and verify using derivative formulas. 2. **Definition of Derivati

1. **Problem:** Differentiate the function $y = 2x^2 + 3x - 20$. 2. **Formula:** The derivative of $x^n$ is given by $\frac{d}{dx} x^n = n x^{n-1}$.

1. **State the problem:** Find the volume of the solid generated by revolving the region bounded by the curves $y=2x-1$, $y=\sqrt{x}$, and $x=0$ about the y-axis using the shell me

1. **Problem statement:** Find the closed form of the power series \(\sum_{n=2}^\infty (n-1) x^{n+6}\) using the power series representation of \(\frac{1}{1-x}\). 2. **Recall the g

1. **State the problem:** Calculate the definite integral $$\int_{-1}^1 (x^5 + 7x^4) \, dx$$. 2. **Recall the integral rules:** The integral of a sum is the sum of the integrals, a

1. **State the problem:** We are given three differentiable functions $f(x)$, $g(x)$, and $h(x)$ where $f(x) = -10x^2 - 6$, the tangent line to $g(x)$ at $x = -2$ is $y = -10x - 8$

1. **Problem Statement:** We are given the function $$h(x) = x^4 - 3x^3 - 10x + 2$$ and need to find its first and second derivatives, determine the critical points by solving $$h'

1. The problem is to find the derivative of the function $g(x) = -6x - x^3$ and understand why the derivative is $g'(x) = -6 - 3x^2$. 2. The formula for the derivative of a functio

1. **Problem:** Calculate the integral $$\int \frac{x^3 - 2x + 4}{x^2 - 1} \, dx.$$\n\n2. **Formula and rules:** When integrating rational functions where the degree of the numerat

1. **Problem Statement:** We analyze the function $$f(x) = 3 \ln(x + 2) - x$$ to find where it is concave up and concave down, and explain why concave-up regions help gradient desc

1. **Problem statement:** Given $\lim_{x \to c} f(x) = 10$ and $\lim_{x \to c} g(x) = -14$, evaluate the following limits using limit laws. 2. **Limit laws used:**

1. **Problem statement:** Given $$\lim_{x \to c} f(x) = 10$$ and $$\lim_{x \to c} g(x) = -14$$, evaluate $$\lim_{x \to c} (f(x) + 8g(x))$$ using limit laws. 2. **Limit laws used:**

1. The problem is to evaluate the indefinite integral $$\int (5x^3 - 2x + 4) \, dx$$. 2. The formula for integrating a power function is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

1. **Problem:** Find the derivative of $y = e^{\sin 4x}$. 2. **Formula and rules:** The derivative of $e^u$ with respect to $x$ is $e^u \cdot \frac{du}{dx}$ (chain rule).

1. The problem is to find the derivative of a function, but the function is not specified in the question. 2. The derivative of a function $f(x)$ is found using the definition or r

1. **Problem statement:** Given the function $m(q) = 2q e^{-q}$, we need to state the differentiation rules applicable and find the derivative at $q=0$. 2. **Differentiation rules:

1. We are asked to find the limit of the function $f(x)$ as $x$ approaches $-5$. 2. The limit of a function at a point $a$ is the value that $f(x)$ approaches as $x$ gets arbitrari

1. **State the problem:** Evaluate the integral $$\int (x+1)(3x^2+2) \, dx$$. 2. **Expand the integrand:** Use distributive property:

1. **State the problem:** We want to find the integral of the function $\sin^{-1}(\log x)$, which means we want to compute $$\int \sin^{-1}(\log x) \, dx.$$\n\n2. **Recall the form

1. **Problem Statement:** Calculate the integral of the logarithm function $\int \log(x) \, dx$. 2. **Formula and Rules:** We use integration by parts, where $\int u \, dv = uv - \

1. **State the problem:** Evaluate the definite integral $$\int_0^b \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx$$ where the upper limit $b$ is unspecified.