∫ calculus

1. The problem asks to find all intervals where the function $f(x)$ is concave down on the open interval $(-9,9)$. 2. Recall that concavity is determined by the second derivative $

1. Problem 13: Find $$\lim_{t \to 2} \frac{t^3 + 3t^2 - 12t + 4}{t^3 - 4t}$$. 2. Substitute $t=2$ directly:

1. The problem is to find the derivative of the function $f(t) = te^{-2t}$. 2. We will use the product rule for differentiation since the function is a product of two functions: $t

1. **State the problem:** We want to evaluate the integral $$\int y e^y \, dy$$. 2. **Choose a method:** Use integration by parts, where we let:

1. **Problem statement:** We are given several limit problems and a function $f(x) = ax + b - \sqrt{x^2 + 1}$ to analyze. 2. **Inequality bounds:** For $m \leq \frac{1}{2 - \sin x}

1. **Evaluate the limits:** a) \(\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1}\)

1. The problem is to find the derivative of the function $f(x) = e^{-x^2}$.\n\n2. Recall the chain rule for derivatives: if $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)$.\

1. **Find** $\frac{dy}{dx}$ for $$ y = \frac{x}{(x^6 - 2)^3} \cdot \left( \frac{x^2 - 3}{\sqrt{x+1}} \right)^{\frac{4}{3}} \left(2 + \frac{3}{x}\right)^7 \sqrt{x^2 + 2 + x^2} $$

1. The problem is to find the derivative of the function $$f(s) = \sqrt{s^3 + 1}$$. 2. Rewrite the square root as a power: $$f(s) = (s^3 + 1)^{\frac{1}{2}}$$.

1. **Problem 1:** Find $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ for the piecewise function $$f(x) = \begin{cases} \frac{\sqrt{x^2 + x + 2} - 2}{x^2 -1}, & x > 1,

1. The problem states that \(F(x) = \int_{-2}^x f(t) \, dt\) where \(f\) is continuous on \([-2,2]\). 2. By the Fundamental Theorem of Calculus, if \(F(x) = \int_a^x f(t) \, dt\) a

1. **State the problem:** We need to determine whether the limit $$\lim_{x \to 0} \frac{1 - \cos x}{x}$$ exists or not. 2. **Recall the behavior of cosine near 0:** We know that $$

1. **State the problem:** Find the area enclosed between the circle $x^2 + y^2 = 4$, the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$, and the line $x + y = 2$. 2. **Analyze the cur

1. **Evaluate** $$\lim_{x \to 0} \frac{a^x - b^x}{x}$$ Step 1: Recall the exponential limit property: $$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$$.

1. **Problem 1: Limits from the graph of function $f(x)$** (a) Find $\lim_{x \to 1} f(x)$.

1. **State the problem:** We are given the function $f(x) = e^{2x + 3}$ and asked to find its inverse function $f^{-1}(x)$ and then compute the derivative of the inverse, $(f^{-1})

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases}(x - 1) \sqrt[3]{x^2} & x \leq 0 \\ x^2 \arctan\left(\frac{1}{x}\right) & x > 0\end{cases}$$

1. **State the problem:** We want to find the limit $$\lim_{x \to \frac{\pi}{4}} \frac{\tan(2x) - 1}{\sin\left(x - \frac{\pi}{4}\right)}.$$\n\n2. **Evaluate the numerator and denom

1. **State the problem:** Find the derivative with respect to $x$ of the function $$y = \frac{2x^2 + 2x - 2\ln x}{(x+1)^2}.$$ 2. **Identify numerator and denominator:** Let $$u = 2

1. Evaluate $$\int e^x(1+x)\cos^2(xe^x)\,dx$$. This integral is complex and does not simplify easily with elementary functions; it likely requires advanced techniques or numerical

1. **State the problem:** Simplify the derivative expression $$f'(x) = - \frac{\sqrt{1+x}}{(1+x)^2 \sqrt{1-x}}$$