∫ calculus

1. **State the problem:** Given the function $y = \frac{7}{x}$, find the derivative $\frac{dy}{dx}$. 2. **Recall the formula:** The derivative of $y = x^n$ is $\frac{dy}{dx} = nx^{

1. **Problem Statement:** Given the graph and points, find the values of $f(-2)$, $\lim_{x \to 0^-} f(x)$, $\lim_{x \to 0^+} f(x)$, determine if $\lim_{x \to 0} f(x)$ exists, and f

1. **Problem:** Determine the convergence or divergence of the series \(\sum \frac{1}{n \ln(n)}\) using the Comparison Test. **Step 1:** For all \(n > 1\), note that \(\frac{1}{n \

1. The problem involves determining the convergence or divergence of series using the Comparison Test. 2. The Comparison Test states: If $0 \leq a_n \leq b_n$ for all $n$ beyond so

1. **Problem:** Find the limit \(\lim_{x \to 5} 8x - 2\). 2. **Formula and rule:** For limits of polynomial or linear functions, the limit as \(x\) approaches a value is simply the

1. **State the problem:** Find the derivative of the function $$f(x) = \frac{5}{2x + 3}$$ using the alternative formula for derivatives: $$f'(x) = \lim_{z \to x} \frac{f(z) - f(x)}

1. **Problem statement:** Find the derivative of the function $w = z + \sqrt{z}$ and evaluate it at $z = 4$. 2. **Formula and rules:** The derivative of $w$ with respect to $z$ is

1. **Problem Statement:** Differentiate the function $y = f(x) = \frac{8}{\sqrt{x - 2}}$ and find the equation of the tangent line at the point $(6,4)$ on the graph. 2. **Recall th

1. Problem (6b): Find the maxima and minima of the function $$f(x) = x^3 - 12x^2 + 36x + 17$$ on the interval $$[1, 10]$$. 2. To find maxima and minima, we use the first derivative

1. آپ نے پوچھا ہے کہ کیا لائن انٹیگرل ملٹیپل انٹیگرل میں ہی آتا ہے۔ 2. لائن انٹیگرل ایک خاص قسم کا انٹیگرل ہے جو کسی ویکٹر فیلڈ یا اسکیلر فیلڈ کے ساتھ کسی کرورڈ لائن پر لیا جاتا ہے

1. The problem is to understand the limit notation $\lim_{x \to a}$ and what it represents in calculus. 2. The limit $\lim_{x \to a} f(x)$ describes the value that the function $f(

1. **Problem Statement:** Verify that each function satisfies Rolle's theorem on the given interval and find all values $c$ where $f'(c) = 0$.

1. **State the problem:** Evaluate the definite integral $$\int_0^1 x^4(1 - x^5) \, dx$$. 2. **Formula and rules:** We use the linearity of integrals and the power rule for integra

1. Bài toán yêu cầu tìm các điểm mà hàm số y = x^2(1 - 2x) không đạt cực trị. 2. Để tìm cực trị của hàm số, ta cần tính đạo hàm y' và giải phương trình y' = 0.

1. The problem is to evaluate the definite integral by applying the appropriate limits. 2. The general formula for a definite integral from $a$ to $b$ of a function $f(x)$ is:

1. The problem is to change the order of integration in the triple integral from $r \, dz \, dr \, d\theta$ to $r \, dr \, dz \, d\theta$. 2. The original integral is expressed as

1. **Problem Statement:** We are given two differentiable functions $f(x)$ and $g(x)$ on the interval $[a,b]$. We want to determine which of the following functions is always incre

1. **State the problem:** Find the derivative $y'$ of the function $y = \tan^{-1}(x^3)$.\n\n2. **Recall the formula:** The derivative of $y = \tan^{-1}(u)$ with respect to $x$ is g

1. **State the problem:** Find the derivative $y'$ of the function $y = \tan^{-1}(x^3)$.\n\n2. **Recall the formula:** The derivative of $\tan^{-1}(u)$ with respect to $x$ is given

1. The problem is to find the derivative $y'$ of the function $y = \tan^{-1}(3x)$.\n\n2. We use the formula for the derivative of the inverse tangent function: $$\frac{d}{dx} \tan^

1. **State the problem:** We are given the function $f(x) = 4 \sin x + 7 \cos x$ and need to find its derivative $f'(x)$ and then evaluate $f'(x)$ at $x = \frac{5\pi}{6}$. 2. **Rec