∫ calculus

1. **Stating the problem:** We are given the function $$f(x) = 2x + \ln(x^2 - 3)$$ and asked to analyze it by finding: a) The domain of $$f$$

1. **State the problem:** Find the value of the limit $$\lim_{x \to 4} (2x - 3)$$. 2. **Formula and rule:** For limits of polynomial or linear functions, the limit as $x$ approache

1. The problem asks about the behavior of the function $f$ at a point $x=a$ where the graph has a break or jump. 2. The statement is: "If the graph of the function $f$ has a break

1. **State the problem:** We need to evaluate the limit of the function $$\frac{x^2 - 4x + 3}{x - 1}$$ as $$x$$ approaches 6. 2. **Recall the formula and rules:** The limit of a ra

1. **State the problem:** We are given two functions $f(x) = x - 3$ and $g(x) = 5$. We need to evaluate the limit $$\lim_{x \to 2} 3g(x).$$ 2. **Recall the limit properties:** The

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1^-} \frac{x^2 - 4x + 3}{x - 1}$$ as $x$ approaches 1 from the left. 2. **Recall the formula and rules:** The limit of a

1. **State the problem:** We need to evaluate the limit $$\lim_{x \to 6^-} \frac{x^2 - 4x + 3}{x - 1}$$ which means finding the value the expression approaches as $x$ approaches 6

1. **State the problem:** Find the derivative of the function $r = es$ with respect to $s$. 2. **Recall the formula:** The derivative of a function $r(s)$ with respect to $s$ is de

1. **State the problem:** Evaluate the limit $$\lim_{x \to 2^-} \frac{x^2 - 4x + 3}{x - 1}$$. 2. **Recall the formula and rules:** The limit of a rational function as $x$ approache

1. **State the problem:** We need to evaluate the limit $$\lim_{x \to 2^-} \frac{x^2 - 4x + 3}{x - 1}$$ which means finding the value the expression approaches as $x$ approaches 2

1. **State the problem:** Find the value of the limit $$\lim_{x \to 2^+} (2x - 3)$$ which means we want to find the value of the expression $2x - 3$ as $x$ approaches 2 from the ri

1. The problem asks about the continuity of a function $f$ at a point $x=a$ where the graph has a break or jump. 2. The definition of continuity at $x=a$ is that the limit of $f(x)

1. **State the problem:** Find the derivative of the function $$y = (x^{9.6})^{\frac{1}{3}} + 2e^{1.3}$$ with respect to $$x$$. 2. **Simplify the function:** Use the power of a pow

1. **State the problem:** Find the derivative of the function $$y = 23x^{9.6} + 2e^{1.3x}$$ with respect to $$x$$. 2. **Recall the formulas:**

1. **Problem statement:** Find the area of the region bounded by the curves $y = x^2 + 2$, $y = 6 - x^2$, and the line $y = 3$.

1. **State the problem:** Find the area under the curve $y = 3x^2 - 4x$ bounded by the vertical lines $x = -1$, $x = 2$, and the $x$-axis. 2. **Understand the problem:** The area u

1. The problem is to find the derivative of the function $y = 3x^2 + 5x - 7$. 2. The formula for the derivative of a polynomial function $f(x) = ax^n$ is $f'(x) = n \cdot a x^{n-1}

1. **State the problem:** We need to find $\frac{dy}{dx}$ using implicit differentiation for the equation $$3y^2 + \tan^3 x = x^2 + 4xy.$$\n\n2. **Recall the formula and rules:** I

1. **Problem Statement:** Sketch a graph of a function $f$ such that:

1. **Problem Statement:** Find the slope of the secant line PQ where $P=(1,0)$ and $Q=(x, \sin(\frac{10\pi}{x}))$ for given values of $x$. Then analyze if these slopes approach a l

1. **Problem Statement:** Determine the values of $a$ for which $\lim_{x \to a} f(x)$ exists for the piecewise function: $$f(x) = \begin{cases} e^x & \text{if } x \leq 0 \\ x - 1 &