∫ calculus

1. The problem is to understand the concept of derivability (differentiability) of a function. 2. Derivability means that a function has a derivative at a given point, which repres

1. **State the problem:** Find the point on the curve $y = -2x^4$ where the tangent line is perpendicular to the line $x - y + 1 = 0$. 2. **Rewrite the given line in slope-intercep

1. **State the problem:** We are given the variables $x$ and $y$ defined in terms of $u$ and $v$ as: $$x = \frac{u}{v}, \quad y = \frac{1}{v - u}$$

1. **State the problem:** We are given the marginal cost function $C'(q) = 500 + 3q + q^2$ and asked to find the value of $C'(5)$, which represents the marginal cost when producing

1. **State the problem:** We are given the cost function $$C(q) = 500 + 3q + q^2$$ and asked to find the second derivative $$C''(5)$$. 2. **Recall the formula:** The first derivati

1. **Problem statement:** Given the function $f(x) = x^2 \ln(3 + 2x)$, we want to find the coefficients $c_n$ of the power series representation of its integral $\int f(x) \, dx =

1. **State the problem:** We are given the power series $$\sum_{n=1}^{\infty} \frac{(x-12)^n}{5^n \sqrt[3]{n}}$$ and need to find its radius of convergence $R$, and the interval en

1. **Problem statement:** Sketch the graph of the function $y$ given that $y(1) = 0$ and the graph is a parabola opening upwards with vertex below the x-axis, crossing the x-axis n

1. **State the problem:** We have the curve $$y = 2x + 8 - \frac{5}{x^2}$$ for $$x > 0$$.

1. **State the problem:** We want to approximate the value of $\frac{1}{12} \int_0^{12} M(t) \, dt$ using a left Riemann sum with 4 subintervals based on the given table values of

1. **State the problem:** We want to approximate the average amount of water in the tank over the interval $[0,12]$ using a right Riemann sum with 4 subintervals. 2. **Given data:*

1. The problem states that the graph of the second derivative $f''$ of a continuous function $f$ is given on the interval $[0,9]$, and $f$ has only one critical point at $x=6$. 2.

1. **State the problem:** Determine for which function the Second Derivative Test cannot conclude that $x=2$ is a relative minimum or maximum. 2. **Recall the Second Derivative Tes

1. **State the problem:** Determine for which function the Second Derivative Test cannot conclude if $x=2$ is a relative minimum or maximum. 2. **Recall the Second Derivative Test:

1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = e^x \csc(2y) \csc(y)$$ with initial condition $$y(0) = \frac{\pi}{6}$$. 2. **Rewrite the equation:** Sep

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = e^{x} \csc 2y \csc y$$ with the initial condition $$y(0) = \frac{\pi}{6}$$.

1. **State the problem:** Find the partial derivative of the function $f(x,y) = x^2 y + \cos(x)$ with respect to $x$. 2. **Recall the formula:** The partial derivative of $f(x,y)$

1. **State the problem:** We have a curve $C$ defined parametrically by $x = t^2 + t$ and $y = 4t^2 - t^3$ for $t \geq 0$. We need to find the gradient of $C$ at the point where it

1. We are asked to find the integral $$\int x \sin(mx) \, dx$$ where $$m \neq 0$$. 2. The formula for integration by parts is $$\int u \, dv = uv - \int v \, du$$.

1. **Stating the problem:** We want to find the integral $$\int \sin(px) \cos(qx) \, dx$$ where $p \neq q$ and $p \neq -q$. 2. **Formula and important rule:** Use the product-to-su

1. **State the problem:** We analyze the function $$f(x) = (2x + 1)e^{1 - 2x}$$ and find its derivative, limit behavior, tangent line at $$x_0 = \frac{1}{2}$$, and asymptotic prope