∫ calculus

1. **State the problem:** Find the second derivative of the function $$f(x)=\frac{4e^{2x}+3x}{e^{x^{2}}}.$$\n\n2. **Rewrite the function:** Simplify the expression by writing it as

1. **State the problem:** Differentiate the function $$f(x) = \frac{e^{x^3} - 1}{x}$$ with respect to $$x$$. 2. **Formula used:** We will use the quotient rule for differentiation,

1. **State the problem:** We are given a piecewise function $$f(x) = \begin{cases} x^3 + 2, & x > 0 \\ 3 - 2x^2, & x < 0 \end{cases}$$

1. **State the problem:** Find the equation of the tangent line to the curve of the function $f(x) = e^{2x+1}$ at the point $\left(-\frac{1}{2}, 1\right)$. 2. **Recall the formula

1. We are given the function $f(x) = e^x \cdot g(x)$, with $g(0) = 2$ and $g'(0) = 1$. We need to find $f'(0)$. 2. To find the derivative of a product of two functions, we use the

1. **State the problem:** Find the derivative $f'(x)$ of the function $f(x) = \csc x \cot x$. 2. **Recall the formulas and rules:**

1. **Stating the problem:** Find the area inside both polar curves $r=3+2\cos(\theta)$ and $r=3+2\sin(\theta)$.\n\n2. **Formula for area inside a polar curve:** The area enclosed b

1. **State the problem:** We need to find the limit $$\lim_{x \to -3} (f(x) + h(x))$$ where functions $f$ and $h$ are given graphically. 2. **Recall the limit sum rule:** The limit

1. **State the problem:** We need to find the limit $$\lim_{x \to 3} \frac{g(x)}{f(x)}$$ where functions $g$ and $f$ are given graphically. 2. **Analyze the behavior of $g(x)$ near

1. **State the problem:** Find the limit $$\lim_{x\to -1} \frac{12x^3 + 12x^2}{x^4 - x^2}$$. 2. **Analyze the expression:** The limit is a rational function. We first check if dire

1. **State the problem:** We need to find the limit $$\lim_{x \to \frac{\pi}{4}} \frac{\cos(2x)}{\cos(x) - \sin(x)}$$. 2. **Recall formulas and rules:**

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \frac{x-3}{2-\sqrt{x+1}}.$$\n\n2. **Identify the issue:** Direct substitution gives $$\frac{3-3}{2-\sqrt{3+1}}=\

1. **Problem Statement:** We are given a piecewise function: $$f(x)=\begin{cases} \frac{2}{x^2} & \text{for } x \leq -1 \\\ \frac{x+3}{\cos(x+1)} & \text{for } -1 < x < \frac{\pi -

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{|x-1|}{1-x}$$. 2. **Recall the definition of absolute value:**

1. **State the problem:** We need to find the limit $$\lim_{\theta \to 0} \frac{1 - \cos(\theta)}{2 \sin^2(\theta)}$$. 2. **Recall important formulas and rules:**

1. **State the problem:** Find the limit $$\lim_{x\to 1} \frac{2x}{x^2 - 7x + 6}$$. 2. **Recall the formula and approach:** To find the limit of a rational function as $x$ approach

1. The problem asks if Hayley's suggestion to use the functions \(g(x)=e^x\) and \(h(x)=e^{-x}\) to apply the squeeze theorem for function \(f(x)\) near \(x=0\) is correct. 2. The

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \frac{\sqrt{2x-5}-1}{x-3}.$$\n\n2. **Identify the indeterminate form:** Substituting $x=3$ directly gives $$\fra

1. **State the problem:** We are given a piecewise function $$f(x)=\begin{cases}\ln(-x)+3 & \text{for } x < -3 \\\\ \ln(-x+3) & \text{for } -3 \leq x < 3 \end{cases}$$

1. **State the problem:** We need to find the limit $$\lim_{x\to 2} \frac{x^4 + 3x^3 - 10x^2}{x^2 - 2x}.$$\n\n2. **Check direct substitution:** Substitute $x=2$ into numerator and

1. The problem asks which of the functions \(g(x) = \sqrt[5]{x}\) and \(h(x) = \sqrt[3]{x}\) are continuous for all real numbers. 2. Recall that the \(n\)-th root function \(f(x) =