∫ calculus

1. **State the problem:** Find the values of $a$ and $b$ that make the piecewise function $$

1. **Problem statement:** Find the function $f(x)$ given that the slope of the tangent line to $f$ at $x=c$ is given by the limit $$\lim_{h \to 0} \frac{\tan\left(\frac{\pi}{4} + h

1. **Problem statement:** Find the derivative of the velocity function $$v(t) = (t - 5)(t - 2)^2$$ for time $$t \geq 0$$. 2. **Formula used:** To differentiate a product of two fun

1. **State the problem:** We have the position function of an object given by $$s(t) = 6 + 8t - t^2$$ for $$0 \leq t \leq 5$$. We need to determine when the object is moving forwar

1. **State the problem:** Find the second derivative $y''$ when $y = e^{\sqrt{x}}$. 2. **Recall the formula:** To find $y''$, we first find $y'$ using the chain rule, then differen

1. The problem asks to analyze the functions given in problems 12 and 13, focusing on finding inflection points (Wendepunkte) and other characteristics. 2. For problem 12, the func

1. **State the problem:** Differentiate the function $P = 6t^3 + 2e^t$ with respect to $t$. 2. **Recall the differentiation rules:**

1. **State the problem:** Differentiate the function $$P = 6t^3 + 2e^t$$ with respect to $$t$$. 2. **Recall the differentiation rules:**

1. **State the problem:** Find the arc length of the curve defined by the function $y = x^{\frac{3}{2}}$ from the point $(0,0)$ to $(4,8)$. 2. **Formula for arc length:** The arc l

1. **State the problem:** Find the arc length of the curve given by $$y = \frac{x^3}{6} + \frac{1}{2x}$$

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:*