∫ calculus

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 $$f'(x) = - \frac{\sqrt{1+x}}{(1+x)^2 \sqrt{1-x}}$$

1. The problem is to simplify the expression for the derivative: $$f'(x) = - \frac{(1+x)^2}{1-x} \cdot \frac{1+x}{1+x}$$

1. The problem asks to find the value of $c$ in Lagrange's Mean Value Theorem (MVT) for the function $f(x) = x(x - 1)$ on the interval $[1, 2]$. 2. Recall that Lagrange's MVT state

1. The problem is to find the derivative of the natural logarithm of a function $f(x)$ with respect to $x$. 2. Recall the chain rule for derivatives: if $y = \ln(f(x))$, then the d

1. Problem 108: Find $f'(x)$ for $f(x) = (x^2 + 1)(x^3 + 3)$ in two ways. (a) Multiply first, then differentiate:

1. **State the problem:** We are given the function $$f(x) = -2\sqrt{x^3} - \sqrt{x}$$ and need to find its derivative at $$x=3$$, i.e., $$f'(3)$$. 2. **Rewrite the function using

1. **State the problem:** We are given the function $$f(x) = \frac{5\sqrt{x}}{3} + 2\sqrt{x^3}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. We will express the an

1. **State the problem:** We are given the function $$f(x) = \frac{5\sqrt{x}}{3} + 2\sqrt{x^3}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. We will express the an

1. **State the problem:** We are given the function $$f(x) = \frac{2}{x} - \frac{1}{2x^2}$$ and need to find the derivative at $$x=2$$, i.e., $$f'(2)$$. 2. **Rewrite the function:*

1. **State the problem:** We are given the function $$f(x) = -\frac{2 \sqrt{x}}{5} + \frac{2 \sqrt{x^3}}{3}$$ and need to find its derivative at $$x=4$$, i.e., $$f'(4)$$. 2. **Rewr