∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{\ln\left(x - \sqrt[3]{2x} - 3\right)}{\sin\left(\frac{\pi x}{2}\right) - \sin\left(\pi(x - 1)\right)}$$

1. **Problem Statement:** Given the function $d(s) = 3^1$ (a constant function), find the limit as $s$ approaches any value $a$.

1. The problem is to find the derivative $\frac{du}{d\theta}$ given $u = \sin^2 \theta$.\n\n2. Recall the chain rule for derivatives: if $u = f(g(\theta))$, then $\frac{du}{d\theta

1. Evaluate $$\int \cos x \ e^{\sin x} \, dx$$ Use substitution: let $$u = \sin x$$, then $$du = \cos x \, dx$$.

1. **State the problem:** We are given a piecewise function $$f(x) = \begin{cases} 3, & x \leq -1 \\ (x-1)^2, & x > -1 \end{cases}$$

1. **Problem Statement:** Evaluate the limit of the function $f(x)$ as $x$ approaches 3 given the graph. 2. **Understanding the Graph:** The graph is a straight line passing throug

1. The problem is to find the derivative of the function $y=(2x+3)^5$ using the chain rule. 2. The chain rule states that if $y=f(g(x))$, then the derivative is $\frac{dy}{dx} = f'

1. **State the problem:** Evaluate the integral $$\int \frac{\tan^{-1} \left( \frac{1}{x} \right)}{1 + x^2} \, dx$$.

1. We are asked to evaluate the definite integral $$\int_0^3 (x^2 + 4) \, dx$$ which means finding the area under the curve of the function $f(x) = x^2 + 4$ from $x=0$ to $x=3$. 2.

1. **Problem:** Given $x = (2 - y)^5$, find the expression for $5x \frac{dy}{dx} + 2$. 2. **Step 1: Differentiate both sides with respect to $x$.**

1. **State the problem:** We are given the function $f(x) = x^2 - \sqrt{x}$ and need to find the equation of the tangent line at $a = 1$. 2. **Recall the formula for the tangent li

1. **State the problem:** Find the limit as $x$ approaches infinity of the expression $$\sqrt{x+1} - \sqrt{x+2}.$$\n\n2. **Recall the formula and technique:** When dealing with lim

1. **State the problem:** Find the derivative of the function $$f(x) = \sqrt[3]{(x+2)^2} (3x - 14)$$ with respect to $$x$$. 2. **Rewrite the function:** Note that $$\sqrt[3]{(x+2)^

1. **State the problem:** Find the limit $$\lim_{x\to 4} \frac{x^3 - 4x^2}{x^2 - 16}$$. 2. **Recall the formula and rules:** To find limits involving rational functions where direc

1. **Problem:** Find the derivative $\frac{dy}{dx}$ if $y = \frac{x^2 + 1}{\sqrt{x}}$. 2. **Formula and rules:** Use the quotient rule or rewrite the function to simplify different

1. **State the problem:** We need to find the limit $$\lim_{x \to 3} \frac{x^3 - 9x}{x^2 - 3x}$$. 2. **Analyze the expression:** Substitute $x=3$ directly:

1. **State the problem:** Find the limit $$\lim_{x\to -1} \frac{12x^3 + 12x^2}{x^4 - x^2}$$. 2. **Check direct substitution:** Substitute $x = -1$:

1. **State the problem:** Find the limit $$\lim_{x\to -4} \frac{x^2 - 15}{x^2 - 16}$$. 2. **Substitute the value directly:** Substitute $x = -4$ into the expression:

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \big(h(x)g(x)\big)$$ where functions $h$ and $g$ are given graphically. 2. **Recall the limit product rule:** If

1. **State the problem:** We have a particle moving along a horizontal axis with position function $s = f(t) = 5t^2 - 2$. We want to find the velocity $v$ at time $t = 10$ seconds.

1. **State the problem:** We are given the function $f(x) = \frac{1}{3}x^3 - x^2$ and need to find its derivative $f'(x)$. 2. **Recall the derivative rules:**