∫ calculus

1. Let's start by stating the problem: Find the derivative of the function $$g(t) = \left(1 + \frac{\sin 3t}{3} - 2t\right)^{-1}$$ with respect to $t$.

1. Problem 1: Find the derivative of $$f(x) = (2x^3 - 8x^2 + 5)^4$$. 2. We apply the chain rule: If $$f(x) = [u(x)]^4$$, then $$f'(x) = 4[u(x)]^3 \cdot u'(x)$$.

1. **Problem 1: Find the derivative of** $y=\cos(\sin(x))$. 2. Let $f(u)=\cos(u)$ and $g(x)=\sin(x)$ so that $y=f(g(x))$.

1. We are given the function $$f(x) = \left(\frac{\sin x}{1 + \cos x}\right)^2$$ and need to find its derivative. 2. Start by using the chain rule: $$f'(x) = 2 \cdot \frac{\sin x}{

1. Stated problem: Evaluate the integral $$\int \frac{1}{\sin y} \, dy$$. 2. Rewrite the integrand using the cosecant function: $$\int \csc y \, dy$$.

1. **State the problem:** We need to evaluate the integral $$\int \frac{1}{\sin y} \, dy.$$ 2. **Rewrite the integral:** Recall that $$\frac{1}{\sin y} = \csc y,$$ so the integral

1. The problem is to express the function $y = 2 + \sqrt{x+4}$ and then rearrange it to express $x$ in terms of $y$ before integrating between the bounds $y = 3$ and $y = 6$. 2. St

1. The problem involves finding the area between the curves defined by $y=6$ and $y=3$. 2. Since both curves are horizontal lines, the area between them over any interval on the x-

1. The problem asks to rearrange the expression $2 + \sqrt{x+4}$ in the form $y = ...$ before integrating. 2. We start by defining the function:

1. The problem asks to integrate the function $x = 4 + \sqrt{y+4}$ with respect to $y$ from $y=3$ to $y=6$. 2. Set up the definite integral:

1. We are asked to find the derivative of the function $$y = (2x - 5)^{-1} (x^2 - 5x)^6$$. 2. Notice this is a product of two functions:

1. Problem (a): Find Taylor polynomials $p_0, p_1, p_2, p_3$ for $f(x)=\ln x$ at $x=1$. - We use Taylor series expansion about $a=1$:

1. We are asked to find the derivative of the function $$y = (4x+3)^4 (x+1)^{-3}$$. 2. This is a product of two functions: $$u = (4x+3)^4$$ and $$v = (x+1)^{-3}$$. We will use the

1. **State the problem:** Determine the truth value of each limit statement about the function $y=f(x)$ based on the graph description. 2. **Analyze each limit:**

1. Stating the problem: Find the derivatives of the functions $$V(t) = \sqrt{14 + 3t}$$, $$f(z) = z^2 + 3$$, and $$W(t) = \frac{1}{\sqrt{t}}$$ using the definition of the derivativ

1. The problem is to analyze the function $$f(x) = 2 + \frac{3}{1+(x+1)^2}$$ and find its critical values, intervals of increase and decrease, global maxima and minima, and limits

1. Stating the problem: Find the derivatives of the functions using the definition of the derivative. 2. For the function $V(t) = \sqrt{14 + 3t}$, the derivative definition is:

1. We are asked to prove that \(\lim_{x \to 1} f(x) = 1\) where \(f(x) = \begin{cases} x^2, & x \neq 1 \\ 2, & x = 1 \end{cases}\). 2. We want to show for every \(\varepsilon > 0\)

1. The problem requires finding $\delta$ for each limit statement given $f(x)$, $x_0$, $L = \lim_{x \to x_0} f(x)$, and $\varepsilon$. We want $\delta > 0$ s.t. if $0 < |x - x_0| <

1. **State the problem:** Show that $$\lim_{n \to \infty} x_n = 1$$ where $$x_n = \frac{n^n}{n!}$$. 2. **Recall Stirling's approximation:** For large $$n$$, $$n! \approx \sqrt{2 \p

1. **Problem Statement:** a) Determine the convergence and interval of convergence for the power series: