∫ calculus

1. **State the problem:** Evaluate the integral of the function composed of three triangles above the x-axis:

1. **State the problem:** We need to find the domain and range of the function $f(x)$ shown in the graph, determine if it is continuous, and identify any local or global extrema. 2

1. **Problem:** Find the rate of change of area of a circle with respect to radius $r$ at $r=6$ cm. Formula: Area $A = \pi r^2$.

1. **State the problem:** Find the second derivative of the function $$y = \tan^{-1}(\sqrt{x})$$. 2. **Recall the formula:** The derivative of $$\tan^{-1}(u)$$ with respect to $$x$

1. **State the problem:** We need to find the open intervals where the rate of change of $g(x)$ (i.e., $g'(x)$) is positive and increasing, positive and decreasing, negative and in

1. ปัญหาคือการทำความเข้าใจอินทริเกต (integrals) ทั้งหมด 2. อินทริเกตคือการหาพื้นที่ใต้กราฟของฟังก์ชัน หรือการหาปริพันธ์ของฟังก์ชัน

1. **State the problem:** We want to find the derivative $h'(0)$ of the function $$h(x) = \frac{f(x)}{g(x)}$$ at $x=0$, given values for $f(0)$, $f'(0)$, $g(0)$, and $g'(0)$. 2. **

1. **State the problem:** We are given the revenue function $$R(x)=41x-\frac{x^2}{250}$$ where $x$ is the number of picnic tables sold.

1. **Problem:** Find $\int(13 - 8x + 25x^8)\,dx$. **Step 1:** Use the power rule for integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.

1. **State the problem:** We are given the implicit function $$y^2 - x^2 = 486$$ and the point $$\left(9, 9\sqrt{7}\right)$$ on its graph. We need to find the slope $$m$$ of the ta

1. **State the problem:** We need to find $\frac{dy}{dx}$ for the implicit equation $$x^2y^3 - 8xy^2 = 4$$ using implicit differentiation. 2. **Recall the rule:** When differentiat

1. The problem is to evaluate the function $x(t)$ given by the integral expression: $$x(t) = \int_0^1 x'(s) \, ds + x(0)$$

1. **Problem 5:** Find the Laplace transform of $$f(t) = e^{2t} \sinh(t)$$ 2. **Recall the Laplace transform definition:**

1. **Probleemstelling:** We willen de waarde van de partiële afgeleide $\frac{\partial f}{\partial x}$ van de functie $$f(x,y) = 4e^{1} 2x + 2x \cdot e^{y}$$

1. **State the problem:** We need to find the total area between the curve $$f(x) = -x^2 + 4$$ and the x-axis from $$x = -1$$ to $$x = 3$$. 2. **Understand the curve:** The functio

1. **State the problem:** We want to find the indefinite integral $$\int 12x\sqrt{2x^2-32} \, dx$$. 2. **Recall the formula and substitution:** Let $$u = 2x^2 - 32$$. Then, $$\frac

1. **State the problem:** We are asked to find the rate of change of the average rates of change for the functions given over consecutive equal-length intervals.

1. **State the problem:** We are given the function $f(x)=\dfrac{1}{5x-4}$ and asked to find its derivative $f'(x)$ using the definition of the derivative.

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 c

1. **State the problem:** We are given a line $y = 4x - 6\pi$ that is tangent to the graph of a function $f$ at $x = a$. We need to find which function $f$ and value $a$ from the o

1. **State the problem:** We need to evaluate the double integral $$\int_{x=\frac{\pi}{2}}^{\frac{3\pi}{2}} \int_{y=\cos x}^{\frac{1}{x}} x \, dy \, dx.$$ This means integrating th