∫ calculus

1. **State the problem:** Find the limit as $y$ approaches 3 of the function $$\frac{3x - 2}{1 - 4x}$$. 2. **Clarify the variable:** The expression involves $x$, but the limit is g

1. **Problem 7a:** Find $$\lim_{x\to+\infty} \left(1+\frac{a}{x}\right)^{bx}$$ 2. **Formula and explanation:** This is a classic limit that resembles the definition of the exponent

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ from the implicit equation $$x \tan y - y^2 \ln x = 4.$$\n\n2. **Recall the formula and rules:** Since $y$

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = 8x + \frac{1}{x}$ using the quotient rule. 2. **Recall the quotient rule:** For a function $y = \

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = 8x + \frac{1}{x}$.\n\n2. **Recall the derivative rules:**\n- The derivative of $x^n$ is $nx^{n-1}

1. Problem: Find the limit $$\lim_{x\to+\infty} \left(1+\frac{a}{x}\right)^{bx}$$ Formula: This is a classic limit that resembles the definition of the exponential function $$e^y =

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = \frac{4x + 5}{2x^2 + x}$ using the quotient rule. 2. **Recall the quotient rule formula:** If $y

1. **State the problem:** Find the value of the limit $$\lim_{x \to \frac{\pi}{2}} \frac{7\sqrt{2}(\sin x + \sin 3x)}{2 \sin(2x) \sin\left(\frac{3x}{2}\right) + \cos\left(\frac{5x}

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = (7x + 4x^2)(9 - 9x).$$ 2. **Formula used:** We will use the product rule for derivatives, which

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ where $y = (2x^2 + 1)(x + 1)$.\n\n2. **Formula used:** To differentiate a product of two functions, use the product ru

1. **Problem Statement:** Evaluate each integral of the function $f(x)$ by interpreting the integral as the net area between the graph of $f(x)$ and the $x$-axis over the given int

1. **State the problem:** We need to find the derivative with respect to $x$ of the function $f(x) = (x+2)(x^2+x)$. This requires using the product rule. 2. **Recall the product ru

1. **Problem:** Find the equation of the tangent at the point $(2,2)$ of the curve given by $$xy^2 = 4(4-x).$$ 2. **Formula and rules:** To find the tangent line to a curve at a po

1. **State the problem:** Find all local maxima and minima of the function $$y = x e^{-x^2}$$. 2. **Find the derivative:** Use the product rule for differentiation: $$\frac{d}{dx}[

1. **State the problem:** We need to determine which statements about the global maxima and minima of the function $$f(x) = x^3 + \frac{48}{x^2}$$ on given intervals are true. 2. *

1. **State the problem:** We want to use the Taylor series expansion about $x=0$ to determine whether the function $f(x) = \sin^3(x^3)$ has a local maximum, local minimum, or neith

1. **State the problem:** We want to find the Taylor series expansion of the function $$f(x) = e^{\sin(x^4)} \cos(x^2)$$ about $$x=0$$ and determine the nature of the critical poin

1. Problem Q.3(a): Find $\frac{\partial y}{\partial x}$ if $y = x^3 \ln \sqrt{x}$. 2. Use the property $\ln \sqrt{x} = \frac{1}{2} \ln x$ to rewrite $y = x^3 \cdot \frac{1}{2} \ln

1. **Stating the problem:** We want to find the limit of the function $$f(x) = \frac{1 + \sin x - \cos x + \ln(1 - x)}{x \tan^2 x}$$ as $x$ approaches 0 using L'Hôpital's rule. 2.

1. **Stating the problem:** Simplify and find the limit of the function $$\frac{1+\sin x - \cos x + \ln(1-x)}{x \tan^2 x}$$ as $x \to 0$ using L'Hôpital's rule. 2. **Recall the for

1. **Problem Statement:** Find the limit of a function using a change of variable. 2. **General Idea:** When evaluating limits, sometimes substituting a new variable simplifies the