∫ calculus

1. We need to analyze the continuity of a function to determine at which points it is continuous. 2. A function is continuous at a point $x=a$ if the following three conditions are

1. **State the problem:** We want to find the limit as $y$ approaches 1 of the function $$f(y) = \sec \left(y \sec^3 y - \tan^2 y - 1\right).$$ 2. **Evaluate the expression inside

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x + x^2 + x^3 + \cdots + x^{2025} - 2025}{x^2 - x}.$$\n\n2. **Analyze the numerator:** The numerator is a sum of pow

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x + x^2 + x^3 + \cdots + x^{2025} - 2025}{x^2 - x}$$. 2. **Analyze the numerator:** The numerator is the sum of powe

1. The problem asks to find the limit as $t$ approaches 0 of the function $f(t) = \sin \left( \frac{\pi}{2} \right) \cos(\tan t)$ and to determine the point where the function is c

1. **Stating the problem:** We need to find the solution set for the inequality $f'(x) < 0$. This involves identifying where the derivative of the function $f(x)$ is negative. 2. *

1. **Stating the problem:** We want to evaluate the integral $$\int \frac{\cos(2x)+2\sin^2(x)}{\cos^2(x)}\,dx.$$\n\n2. **Simplify the integrand:** Recall the identity $$\cos(2x) =

1. Find the discontinuity points of the function $f(x)$.\nThe problem states that the function is discontinuous at $x=2$ and $x=3$ because the values of $f(x)$ near these points do

1. The problem asks us to evaluate the definite integral $$\int_a^b f(x)\,dx$$. 2. This integral represents the area under the curve of the function $f(x)$ from $x=a$ to $x=b$.

1. Problem statement: Find the volume of the solid obtained by revolving the region bounded by $x=2\sqrt{y}$ and $y=2\sqrt{x}$ about the x-axis. 2. Convert the equation $x=2\sqrt{y

Problem: Find $\frac{f(a+h)-f(a)}{h}$ and simplify for each given function. 1. For $f(x)=6x-9$.

1. The problem asks us to find the local minimum points of the polynomial function $$g(x) = -4x^4 + 9x^3 + 2x^2 - 7x - 2$$ using the ALEKS graphing calculator, rounding the answers

1. Problem statement: Find the centroid of the planar region bounded by $x=2\sqrt{y}$ and $y=2\sqrt{x}$, which meet at $(0,0)$ and $(4,4)$.\n2. Convert to functions of $x$: $x=2\sq

1. **Problem statement:** Find the centroid of the area bounded by the curves $x=2\sqrt{y}$ and $y=2\sqrt{x}$.\n\n2. **Rewrite curves in terms of \(y\):**\nFrom $x = 2\sqrt{y}$, sq

1. Problem: Find and simplify the difference quotient $$\frac{f(a+h) - f(a)}{h}$$ for the given functions. (i) \( f(x) = 6x - 9 \)

Problem: Compute the difference quotient $\frac{f(a+h)-f(a)}{h}$ for the given functions and simplify. 1. For $f(x)=6x-9$.

1. **State the problem:** We want to find the area enclosed between the curves defined by $$x = 2 \sqrt{y}$$ and $$y = 2 \sqrt{x}$$ which intersect each other. 2. **Rewrite the equ

1. Problem (a): Find the derivative of $$\tan^{-1}(\sqrt{3}x) + (\tan^{-1}x^2)^2$$.\nStep 1: Use chain rule and derivative of arctan: $$\frac{d}{dx}[\tan^{-1}u] = \frac{u'}{1+u^2}$

1. We are asked to find the first derivatives of the following functions: (a) $$f(x) = \tan^{-1}(\sqrt{3x}) + (\tan^{-1}(x^{2}))^{2}$$

1. **State the problem:** Analyze the continuity of the piecewise function $$f(x) = \begin{cases}\frac{x^2 - 1}{x - 1}, & x < 1 \\ \sqrt{x+3} - 2, & x \geq 1\end{cases}$$

1. **State the problem:** We need to find the limit $$\lim_{x \to -\infty} \frac{x^3 + 4x^2}{2x - 1}$$ as $x$ approaches negative infinity. 2. **Analyze the degrees:** The numerato