∫ calculus

1. **Problem Statement:** Find the critical values, intervals of increase and decrease, and relative maxima and minima for the cost function $$C(x) = -0.1x^2 + 18x - 340$$ on the d

1. The problem asks to list the x-values of all relative maxima and minima of the function $$C(x) = -0.1x^2 + 18x - 340$$ on the interval $$0 < x < 400$$. 2. To find relative maxim

1. **State the problem:** We want to find the limit $$\lim_{x \to 0^+} \left(e^x + x\right) \frac{1}{x}$$ as $x$ approaches $0$ from the right. 2. **Recall the limit properties and

1. **State the problem:** Find the limit $$\lim_{x \to 0^+} \left(e^2 + x\right) \frac{1}{x}$$ as $x$ approaches $0$ from the right. 2. **Recall the limit properties:** When evalua

1. **State the problem:** Find the limit $$\lim_{x \to 0^+} \frac{e^x + x - 1}{x}$$. 2. **Recall the formula and rules:** The limit involves an indeterminate form of type $$\frac{0

1. The problem is to solve a limit using L'Hopital's Rule. 2. L'Hopital's Rule states that if the limit of $\frac{f(x)}{g(x)}$ as $x$ approaches a value results in an indeterminate

1. **State the problem:** Evaluate the limit $$\lim_{w \to \infty} \left(\ln w - \sqrt{w}\right)$$ and the equivalent form $$\lim_{w \to \infty} \sqrt{w} \left(\frac{\ln w}{\sqrt{w

1. **State the problem:** We want to find the indefinite integral $$\int \frac{2x^3 + 1}{x^2 + x + 1} \, dx.$$\n\n2. **Understand the problem:** The integrand is a rational functio

1. **Problem statement:** We study the function $f(x) = x(\sqrt{x} - 2)^2$ for $x > 0$. 2. **Limits at infinity:**

1. The problem is to find the slope of a curve or the rate of change of a function without using differentiation. 2. One common method is to use the definition of the derivative as

1. **Problem Statement:** Why is the integral of an odd function over symmetric limits $[-a,a]$ equal to zero? 2. **Definition of an Odd Function:** A function $f(x)$ is odd if it

1. **State the problem:** Evaluate the definite integral $$\int_{-\infty}^{\infty} \frac{x}{(x^2 + 9)^{3/2}} \, dx$$. 2. **Recall the properties:** The integrand is an odd function

1. The problem asks to evaluate the Riemann sum for a given function using right endpoints as sample points. 2. The Riemann sum formula for a function $f(x)$ over an interval $[a,b

1. **Problem Statement:** Evaluate the Riemann sum for the function $f(x)$ on the interval $[1,5]$ using right endpoints as sample points. 2. **Understanding the Riemann Sum:** The

1. **State the problem:** We need to evaluate the definite integral $$\int_0^1 \frac{4 - x}{(x - 2)^2 (x^2 + 4)} \, dx.$$\n\n2. **Identify the integral type:** This is a rational f

1. **Problem:** Find all numbers $c$ in $[-1,3]$ such that $f'(c) = \frac{f(3)-f(-1)}{3-(-1)}$ for $f(x) = x^2 - 2x - 8$ (Rolle's Theorem requires $f(-1) = f(3)$). 2. **Check Rolle

1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{x^2 - 2x + 1}{|x - 1|}$$. 2. **Rewrite the expression:** Notice that the numerator can be factored as $$x

1. Let's analyze the behavior of a function as $x$ approaches 1. 2. To understand this, we need to know the specific function $f(x)$ you are referring to.

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{x^2 - 2x + 1}{|x - 1|}$$. 2. **Rewrite the expression:** Notice that the numerator can be factored:

1. The problem asks to find an expression for the area of the shaded region. 2. To find the area of a region bounded by curves, we use the formula for the area between two function

1. **State the problem:** We want to find the integral $$\int x \ln(x) \, dx$$. 2. **Formula and method:** We will use integration by parts, which states: