∫ calculus

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

1. **State the problem:** We need to find the value of $k$ such that the limit of the function $$f(x) = \begin{cases} x^2 - k, & x < -3 \\ x^2 + 3, & x \geq -3 \end{cases}$$

1. **Problem Statement:** Define the indefinite integral of a vector-valued function and provide an example. 2. **Definition:** The indefinite integral of a vector-valued function

1. **Problem Statement:** We are given the graph of the first derivative $y = f'(x)$ of a function $f(x)$, which is a parabola opening upwards crossing the x-axis at $x=0$. We need

1. The problem states that the given graph is the first derivative $f'(x)$ of a function $f(x)$, and we need to determine which graph could represent the general shape of $f(x)$. 2

1. **State the problem:** Find the derivative of the function $$f(t) = \sqrt[5]{t} - \frac{1}{\sqrt[5]{t}}$$. 2. **Rewrite the function using exponents:** Recall that $$\sqrt[5]{t}

1. The problem is to define the derivative of vector-valued functions and understand how to compute it with examples. 2. A vector-valued function is a function that outputs a vecto