∫ calculus

1. Problem 49: Given $$\lim_{x \to 4} \frac{f(x) - 5}{x - 2} = 1$$, find $$\lim_{x \to 4} f(x)$$. 2. We use the property of limits and continuity. The expression resembles the defi

1. **State the problem:** Find the derivative of the function $$y = \sin^2(\pi t - 2)$$ with respect to $$t$$. 2. **Recall the formula:** To differentiate $$y = [f(t)]^2$$, use the

1. **State the problem:** We need to find the derivative of the function $$y = \sin^2(\text{piet} - 2)$$ with respect to the variable inside the sine function. 2. **Rewrite the fun

1. **Problem Statement:** Find the derivatives using the definition and rules of derivatives for the given functions and analyze differentiability and continuity.

1. **Problem Statement:** Explain why the limits do not exist for the following: 5. $$\lim_{x \to 0} |x|$$

1. **Problem Statement:** Find the surface area generated by revolving the curve about the y-axis for each given function and interval. 2. **Formula:** The surface area $S$ when re

1. Problem: Find $\frac{dy}{dx}$ for $y = (2x + 1)^5$. 2. Write $y = f(u)$ and $u = g(x)$:

1. Let's start by understanding the problem: you want to know when to factorize and when to take the highest power when evaluating limits. 2. When dealing with limits, especially a

1. **State the problem:** We need to evaluate the limit $$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ for each given function and value of $x$. This limit represents the derivative of

1. **Problem 43:** Given the inequalities $$\sqrt{5 - 2x^2} \leq f(x) \leq \sqrt{5 - x^2}$$ for $$-1 \leq x \leq 1$$, find $$\lim_{x \to 0} f(x)$$. 2. **Problem 44:** Given $$2 - x

1. **Stating the problem:** Evaluate the limit $$\lim_{x \to 2^-} \left(2 - \frac{1}{x-2}\right)$$ and explain why it tends to positive infinity, not negative infinity. 2. **Unders

1. **State the problem:** We are given the function $$f(x) = 3e^{1-x} - 2\sin\left(\frac{\pi x}{2}\right) + x$$ and asked to find the inverse value $$f^{-1}(6)$$, which means we wa

1. **State the problem:** We are given the function $$f(x) = 3e^{1-x} - 2\sin(\pi x) + x$$ and asked to find the inverse value $$f^{-1}(6)$$, which means we want to find the value

1. **Problem Statement:** Determine if the function $f(x) = 3e^{1-x} - 2\sin(\pi x) + x$ is one-to-one. 2. **Recall the definition:** A function is one-to-one (injective) if for ev

1. **Stating the problem:** We need to evaluate the double integral $$\int_0^{\frac{\pi}{2}} \int_0^{\sec \theta} r^2 \cos \theta \, dr \, d\theta$$. 2. **Formula and rules:** The

1. **Stating the problem:** We need to evaluate the double integral $$\int_0^{\frac{\pi}{2}} \int_0^{\sec \theta} r^2 \cos \theta \, dr \, d\theta$$. 2. **Understanding the integra

1. Let's first clarify the problem: We are considering a function with a point at $x=2$ that is an open circle (not included in the function). We want to know if this changes any o

1. Let's clarify the problem: An open point on a graph means the point is not included in the function's domain or range at that coordinate. 2. For example, if a function has a hol

1. We are given the function $f(x) = x^2 - 3x + 8$ and need to find: (a) intervals where $f$ is increasing,

1. **State the problem:** Find the limit of the function $f(x)$ as $x$ approaches $2$ from the left, i.e., $\lim_{x \to 2^-} f(x)$. 2. **Understand the graph behavior near $x=2$ fr

1. Problem 51: Evaluate $$\lim_{x \to 0} \sec \left[ e^x + \pi \tan \left( \frac{\pi}{4 \sec x} \right) - 1 \right]$$ 2. Recall that $$\sec y = \frac{1}{\cos y}$$ and the limit of