∫ calculus

1. **State the problem:** Find the first derivative $f'(x)$ and the second derivative $f''(x)$ of the function $$f(x) = \frac{x}{x^2 - 2}.$$\n\n2. **Recall the formula:** For a fun

1. **State the problem:** Differentiate the function $$F(t) = \frac{At^2}{Bt^5 + Ct^9}$$ with respect to $t$. 2. **Recall the quotient rule:** For a function $$\frac{u(t)}{v(t)}$$,

1. **State the problem:** Differentiate the function $$f(z) = (6 - e^z)(6z + e^z)$$ with respect to $$z$$. 2. **Recall the product rule:** For two functions $$u(z)$$ and $$v(z)$$,

1. **State the problem:** Find the derivative $f'(a)$ of the function $$f(x) = \frac{x^2}{x + 20}$$ at the point $a = 5$ using the definition of the derivative: $$f'(a) = \lim_{x \

1. **Problem statement:** Find the slope $m$ of the tangent to the curve $y = 7 + 5x^2 - 2x^3$ at $x = a$.

1. Consider the differential equation $$\frac{dy}{dx} = \frac{1}{2} x (y - 4)^3.$$ We are asked to find the particular solution passing through given points and analyze the slope f

1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = x - 1$$ and asked to analyze it through several parts including slope field, tangent line, and pa

1. **Problem:** Find the general solution to the differential equation $$2x(y+1) - y y' = 0$$. **Step 1:** Rewrite the equation to isolate $$y'$$.

1. **Problem:** Find the general solution to the differential equation $$2x(y + 1) - y y' = 0.$$ 2. **Step 1: Rewrite the equation**

1. **State the problem:** We need to evaluate the definite integral $$\int_0^{\frac{\sqrt{2}}{2}} \frac{-3x^2}{\sqrt{1 - x^2}} \, dx$$

1. **State the problem:** Evaluate the integral $$\int \frac{-9x^2}{\sqrt{4x - x^2}} \, dx.$$\n\n2. **Rewrite the expression under the square root:** Note that $$4x - x^2 = -(x^2 -

1. **State the problem:** Find the derivative of the function $f(x)$. 2. **Recall the derivative definition and rules:** The derivative of a function $f(x)$, denoted $f'(x)$ or $\f

1. **State the problem:** We have a particle moving along the y-axis with velocity function $v(t) = 12t^2 - 24t$ for $t \geq 0$. The initial position is $x(0) = 10$ cm. We need to

1. **Problem statement:** A particle moves along the x-axis with acceleration $a(t) = 12t + 6$. Given velocity at $t=3$ is 36 cm/sec and initial position $x(0) = 4$ cm, find the po

1. **Problem 1:** A particle moves along the x-axis with acceleration $a(t) = 12t + 6$, velocity at $t=3$ is 36 cm/sec, and initial position $x(0) = 4$ cm. Find the position when v

1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = (2 + 6x)^{\frac{5}{x}}$$ and then evaluate the slope and the value of $y$ at $x=1$. 2. **Use log

1. **State the problem:** We are given the function $$y = \left(\frac{x}{7} + \frac{7}{x}\right)^7$$ and asked to:

1. **State the problem:** We have the function $$y = \frac{x^7 + 7x}{7}$$ and need to (a) decompose it into the form $$y = f(u)$$ and $$u = g(x)$$ with non-identity functions, and

1. **State the problem:** Find the total area of the shaded regions between the curves $y=2-x$ and $y=4-x^2$ over the interval where they intersect. 2. **Find the points of interse

1. **State the problem:** We analyze the given cubic-like function to find its relative maximum, relative minimum, intervals of increase and decrease, domain, and range. 2. **Ident

1. **State the problem:** We want to approximate the area under the curve of the function $f(x) = \sqrt{x^3 + 1}$ from $x=0$ to $x=2$ using the Midpoint (Mid-ordinate) Rule with si