∫ calculus

1. **State the problem:** We are given the function $f(x) = x^2 + x$ and asked to find the derivative using the limit definition:

1. **State the problem:** We need to find the four second partial derivatives of the function $$f(x,y) = \sqrt{6 + 3x^2 + 7y^2}$$ which are $$f_{xx}, f_{yy}, f_{xy},$$ and $$f_{yx}

1. **State the problem:** We need to find the four second partial derivatives of the function $$f(x,y) = \sqrt{6 + 3x^2 + 7y^2}$$ which are $$f_{xx}, f_{yy}, f_{xy},$$ and $$f_{yx}

1. **State the problem:** We need to find the four second partial derivatives of the function $$f(x,y) = y^4 \cos 6x$$. 2. **Recall the definitions:** The four second partial deriv

1. **State the problem:** Find the first partial derivatives $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial y}$ of the function $$h(x,y) = (5 + 3y)^x$$. 2. **Recal

1. **State the problem:** Find the first partial derivatives $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial z}$ of the function $$h(x,z) = \sqrt{\frac{9xz}{3x - 4z

1. **State the problem:** Find the derivative of $$H(v) = (v^6 - 5)(v^4 + 4)$$. 2. **Formula used:** Use the product rule for derivatives: $$\frac{d}{dv}[f(v)g(v)] = f'(v)g(v) + f(

1. **State the problem:** We have a rectangle with length $L = 4 + t$ and height $H = t^6$. We want to find the rate of change of the area $A$ with respect to time $t$, i.e., $\fra

1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ at the point $(1,1)$ for the curve given by the implicit equation $$2y^3 - 3y + 3 = 2x^3.$$\n\n2. **Formula

1. Le problème consiste à trouver la dérivée de la fonction $$f(x) = \frac{x+1}{x^2}$$. 2. La formule utilisée est la règle du quotient pour la dérivation :

1. **Problem Statement:** Calculate the limit $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$. 2. **Formula and Rules:** The limit of a function as $x$ approaches a value $a$ is the value

1. **State the problem:** We want to test if the pairs of functions $$Y_1, y_2$$ are functionally dependent using the Jacobian determinant. 2. **Recall the rule:** Two functions $$

1. **State the problem:** We need to find the intervals on $[-9,9]$ where the function $f$ is increasing. 2. **Recall the rule:** A function $f$ is increasing where its derivative

1. **State the problem:** We need to compare the values of the derivative $f'$ at $x = -5$, $x = 0$, and $x = 5$ based on the graph of $f$. 2. **Recall the meaning of $f'(x)$:** Th

### Problem 3.1 1. **State the problem:** Find the point(s) of intersection of the curves $y = \cos x$ and $y = \frac{2x}{\pi} - 1$ on $0 \leq x \leq \pi$, determine which curve is

1. **Problem Statement:** Evaluate the integrals:

1. **Problem Statement:** We have the accumulated response function:

1. **State the problem:** Find the exact area bounded by the curve $$x = g(y) = 4y^2 - y^3$$, the y-axis (i.e., $$x=0$$), and the lines $$y=1$$ and $$y=3$$ using the limit definiti

1. **Problem 1.1a:** Find times when particle changes direction on $0 \leq t \leq 4$ given velocity $$v(t) = t^2 - 4t + 3.$$ 2. **Step 1:** Particle changes direction when velocity

1. **State the problem:** We need to find the limit $$\lim_{t \to 0} \frac{\tan(6t)}{\sin(3t)}$$. 2. **Recall important limits and formulas:**

1. **State the problem:** Find the equation of the tangent line to the curve $y = 2^x$ that passes through the point $(1, 0)$. 2. **Recall the formula for the tangent line:** The t