∫ calculus

1. **Problem statement:** Why does $x^{1/x}$ get close to $1$ when $x$ gets very big? 2. **Think of it as a tiny power.**

1. **Problem statement:** Explain why $\lim_{x\to\infty}x^{1/x}=1$. 2. **Use logarithms to simplify the exponent.**

1. **Problem statement:** Evaluate $$\lim_{x\to\infty}\left(\frac{4^x+9^x}{x}\right)^{1/x}.$$ 2. **Use the dominant term rule.**

1. **Problem statement:** Evaluate $\lim_{x\to\infty}\left(4^x+9^x\right)^{1/x}$. 2. **Use the dominant-term rule:** For large $x$, the bigger exponential term grows fastest. Since

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty}\left(\frac{4^x + 9^x}{x}\right)^{\frac{1}{x}}.$$\n\n2. **Analyze the expression inside the limit:** The ex

1. **State the problem:** We need to find the volume of the solid formed by rotating the region enclosed by the curves $y = \sqrt{x}$ and $y = 2x - 1$ around the y-axis. 2. **Find

1. **State the problem:** We want to estimate how much farther Kelly travels than Chris during the first 10 seconds using the midpoint rule on their velocity data. 2. **Recall the

1. **Problem statement:** Estimate the area under $f(x)=x^2+1$ on $[0,6]$ using $3$ rectangles of equal width with **right endpoints**, and decide whether the estimate is an overes

1. **Problem statement:** We want to find the net signed area between the graph of $f(x)$ and the x-axis on the given intervals using sigma notation and summation formulas. The rig

1. **State the problem:** Differentiate the function $$f(x) = \frac{\sin 4x}{x^3}$$ with respect to $x$. 2. **Formula used:** We use the quotient rule for differentiation, which st

1. **State the problem:** Differentiate the function $$g(x) = e^x \cdot (e^x - 2)^{-2}$$ with respect to $$x$$. 2. **Recall the product rule:** If $$g(x) = u(x) \cdot v(x)$$, then

1. **Problem Statement:** Evaluate the left-hand limit, right-hand limit, and two-sided limit of the function $f(x)$ at $x = -3, -1, 1, 4$ using the graph. Also, find the function

1. **Problem Statement:** Calculate the integral $$\int (3x^2 + 2x + 1) \, dx$$. 2. **Formula Used:** The integral of a polynomial term $$ax^n$$ is given by $$\int ax^n \, dx = \fr

1. **Problem Statement:** Verify that the derivative of $-\cos x$ is $\sin x$. 2. **Formula and Theory:** The derivative of $\cos x$ with respect to $x$ is $-\sin x$. Using the con

1. **Problem Statement:** Find the integral of $\sin x$ with respect to $x$, i.e., compute $\int \sin x \, dx$. 2. **Formula and Theory:** The integral of $\sin x$ is based on the

1. **State the problem:** We have the position function $$s(t) = -16 t^2 + 104 t$$ and need to find the average velocity over several time intervals starting at $t=1$.

1. **State the problem:** We need to evaluate the integral $$\int \sqrt{1 + y^2} \, dy$$. 2. **Formula and approach:** This is an integral involving a square root of a quadratic ex

1. **Problem Statement:** Calculate the integral $$\int (3x^2 + 2x + 1) \, dx$$. 2. **Formula Used:** The integral of a polynomial term $$ax^n$$ is given by $$\int ax^n \, dx = \fr

1. **State the problem:** Find the limit as $x$ approaches infinity of the expression $$\frac{\sqrt{13x^2 + 5x + 1}}{2x - 3}.$$\n\n2. **Recall the formula and rules:** When dealing

1. **State the problem:** We have a continuous function $f(x)$ on $[-9,9]$ with $f(-3) = -2$ and $f(2) = -7$. The graph between $x=-3$ and $x=2$ is hidden. We want to determine if

1. **Problem statement:** We are given inflow rate functions $f_1(x)$, $f_2(x)$, and $f_3(x)$ describing the rate of water flowing into a tank over time $x$. The tank is initially