∫ calculus

1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{1 - \sqrt{\cos(x)}}{x^2}.$$\n\n2. **Recall relevant formulas and expansions:** For small $x$, use the Taylor series

1. **Problem statement:** Evaluate the improper integral $$\int_0^\infty x \sin x \, dx$$. 2. **Formula and approach:** We use integration by parts for integrals of the form $$\int

1. Vamos resolver a integral (a) $$\int \cos^5 x \sin^4 x \, dx$$. 2. Para integrais de potências de funções trigonométricas, usamos identidades trigonométricas e substituições.

1. **State the problem:** Find the average vertical height of the shaded area between the curves $y = x^2$ and $y = 6 - x$ from $x=0$ to their intersection point. 2. **Find the int

1. **State the problem:** Calculate the definite integral $$\int_0^5 \sqrt{x} \, dx$$. 2. **Recall the formula:** The integral of $$\sqrt{x}$$, which is $$x^{1/2}$$, is given by

1. **Problem Statement:** We approximate the integral $$\int_1^5 \sqrt{x} \, dx$$ using 100 rectangles and right-hand sums. 2. **Step 1: Define the interval and width of each recta

1. **State the problem:** We want to find the limit $$\lim_{x \to 0} x \sqrt{\cos \sqrt{x}}.$$\n\n2. **Recall the limit and approximation rules:** As $x \to 0$, $\sqrt{x} \to 0$. W

1. Problem: Find the derivatives of the given functions. (a) Given $y = \frac{9 - 15x + 4^3 - 7x^5}{x^4}$.

1. **Problem:** Estimate the roots of the equation $$\cos x - 2x^2 = 0$$ using the Maclaurin polynomial. 2. **Recall the Maclaurin series for $$\cos x$$:**

1. **State the problem:** We need to find the indefinite integral of the function $3x^2 + 7x - 2$ with respect to $x$. 2. **Recall the formula for integration:** The integral of a

1. The problem is to find the value of $\delta$ such that if $0 < |x - a| < \delta$ then $|f(x) - L| < \varepsilon$ for the function $f(x) = 8 - x^2$ with $a = 2$, $L = 4$, and $\v

1. **State the problem:** We need to find the value of $W(0) - W(1)$ where $$W(x) = \int \frac{60x}{(3x^2 - 2)^2} \, dx.$$

1. **Stating the problem:** Maclaurin series are a special case of Taylor series centered at $x=0$. They allow us to approximate functions as infinite sums of powers of $x$. 2. **F

1. **State the problem:** We are given the function $$f(x) = \left(\frac{4}{3}x + \frac{5}{4}\right) e^{-\frac{7}{6x+4}}$$ and asked to find its oblique and vertical asymptotes. 2.

1. **Problem Statement:** We are given three graphs and need to determine which corresponds to the function $f$, its first derivative $f'$, and its second derivative $f''$. 2. **Ke

1. **Problem Statement:** We are given the graph of $f'$, the derivative of a function $f$, and need to determine the intervals on the closed interval $[-9,9]$ where $f$ is decreas

1. **State the problem:** Find the second derivative of the function $$h(t) = (t^2 + 1) \sin t$$. 2. **Recall the product rule:** For two functions $u(t)$ and $v(t)$, the derivativ

1. **State the problem:** Find the derivative of the function $f(x) = \tan^2 x$. 2. **Recall the formula:** The derivative of $\tan x$ is $\sec^2 x$. For a function $g(x)^2$, the d

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{x - \sin x}{1 + \cos x}$$. 2. **Recall the formula:** To differentiate a quotient $$\frac{u}{v}$$, use

1. **Problem a:** Find the book value of the tractor at the end of 2 years. Given the problem does not provide an explicit formula for the tractor's book value, we assume a depreci

1. **Problem Statement:** Find the intercepts, asymptotes, and intervals of monotonicity for the function