∫ calculus

1. **State the problem:** Differentiate the function $$f(x) = \frac{x^2 \sin(x) - 23 \log(x^2)}{\sqrt{x}}$$ with respect to $$x$$. 2. **Rewrite the function:** To simplify differen

1. Problem: Evaluate the integral $$\int xe^{2x} \, dx$$ using integration by parts. Formula: Integration by parts states $$\int u \, dv = uv - \int v \, du$$.

1. **Stating the problem:** We want to understand why the derivative of a function $f$ at a point $x$, denoted $f'(x)$, is defined as the limit $$f'(x) = \lim_{h \to 0} \frac{f(x+h

1. **Problem Statement:** Differentiate the function $f(x) = \frac{x}{x-1}$ using the definition of the derivative. 2. **Definition of Derivative:** The derivative $f'(x)$ is defin

1. **Evaluate** $\int_0^2 (3x^2 - 2x + 1) \, dx$. Formula: $\int (ax^n) dx = \frac{a}{n+1} x^{n+1} + C$.

1. **Problem Statement:** Find the volume of the solid under the surface defined by the function $f(x,y) = 7$ and above the region $D$ bounded by the cardioid $r = 1 + \cos(\theta)

1. **Problem:** Find the volume of the solid obtained by rotating the region bounded by $y = 2 - \frac{1}{2}x$, $y=0$, $x=1$, and $x=2$ about the x-axis. 2. **Formula:** For rotati

1. **Problem statement:** Differentiate the following functions: (i) $f(x) = 4x^5 - 5x^4$

1. The problem asks to interpret $\frac{dy}{dx}$ for the implicit function given by $\sin^{-1}(x + y) = \cos^{-1}(xy)$.\n\n2. $\frac{dy}{dx}$ represents the derivative of $y$ with

1. **State the problem:** Find the area enclosed between the curves $y = e^x$ and $y = x^2 + 1$. 2. **Formula and approach:** The area between two curves $y = f(x)$ and $y = g(x)$

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{(1 + mx)^n - (1 + nx)^m}{x^2}$$ without using L'Hôpital's rule.

1. Evaluate $$\int_0^2 (3x^2 - 2x + 1) \, dx$$ - Use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

1. **State the problem:** We want to approximate the area under the curve of the function $f(x) = x^3$ on the interval $[0, 2]$ using the midpoint Riemann sum with 100 subintervals

1. The problem asks why the Riemann sum becomes exact as $n \to \infty$. 2. The Riemann sum approximates the area under a curve by dividing it into $n$ rectangles and summing their

1. Let's first clarify the problem: you want to find the left, right, and midpoint values for $q_6$ in a numerical method context, likely related to Riemann sums or numerical integ

1. **Problem Statement:** Estimate the area under the curve of the function $f$ given by the table:

1. **Problem Statement:** Find the derivative of the function $y = 3x^2 + 5x - 7$. 2. **Formula Used:** The derivative of a function $f(x)$ with respect to $x$ is given by $\frac{d

1. The problem asks to find the partial derivative of the function $u = x + y + z$ with respect to $x$. 2. The formula for the partial derivative of a function $u$ with respect to

1. The problem is to find the Maclaurin series expansion of the function $e^x$ up to the 3rd degree term. 2. The Maclaurin series for a function $f(x)$ is given by:

1. **Stating the problem:** We want to find the values of $\theta$ for which the tangent to the cardioid given by the polar equation $$r = a(1 + \cos \theta)$$ is parallel to the i

1. **State the problem:** We want to find the right Riemann sum for the function $f(x) = 3x + 1$ on the interval $[2, 5]$ using $n$ subintervals, and then compute the limit as $n \