∫ calculus

1. The problem is to understand and apply the basic integral formulas given: - $\int k \, dx = kx + C$, where $k$ is a constant.

1. **Problem A:** Find the area bounded by $y = 4x^3 - x^5$ and the x-axis. 2. Set $y=0$ to find intersection points:

1. The statement "A function can have a relative minimum at a point where its derivative is undefined" is **True**. For example, the function $f(x) = |x|$ has a relative minimum at

1. The problem is to find the volume of a solid of revolution using the cylindrical shell method. 2. Suppose we revolve the region bounded by a function $y=f(x)$, the x-axis, and v

1. Problem: Find the value of $f'(3)$ for $f(x)=x^5+6x+4$. 2. Differentiate $f(x)$:

1. **Problem A:** Find the area bounded by the curve $y = 4x^3 - x^5$ and the x-axis. 2. Set $y=0$ to find the x-intercepts:

1. **Problem 1:** Evaluate the integral for $t_{avg}^2 = \int_1^4 t (\sqrt{x} - \frac{1}{x}) \, dt$. 2. The integral expression and evaluation given are inconsistent and contain er

1. **State the problem:** Find the second order partial derivatives of the function $$f(x,y) = \frac{x - y}{x^2 - y^2}$$ with respect to $x$ and $y$. 2. **Simplify the function:**

1. **State the problem:** Find the second order partial derivatives of the function $$f(x,y) = \frac{x - y}{x^2 - y^2}$$. 2. **Simplify the function:** Note that the denominator ca

1. Masalah: Hitung luas daerah yang diarsir antara kurva $y = x^2$ dan garis $y = 4$ dari $x = 0$ sampai $x = 2$. 2. Daerah yang diarsir adalah area di antara garis horizontal $y =

1. Masalah pertama: Hitung luas daerah di bawah kurva $y = x^2$ dari $x=0$ sampai $x=2$ dan di atas $y=0$ hingga $y=4$. 2. Luas daerah ini adalah integral dari $y = x^2$ dari 0 sam

1. **Problem Statement:** We need to find the reduction formula for the integral $$I_m = \int \cos^m x \, dx$$ and show that

1. **State the problem:** We have the function $$f(x) = \frac{x^2 - 15}{x - 4}$$ and need to find where it is increasing, decreasing, and its local extrema. 2. **Find the derivativ

1. **State the problem:** We have the function $$f(x) = 2x^3 - 12x^2 + 18x - 8$$ and need to find where it is increasing, decreasing, and its local extrema. 2. **Find the derivativ

1. **State the problem:** Find the absolute maximum and minimum values of the function $$f(t) = t - \sqrt[3]{t}$$ on the interval $$[-1,7]$$. 2. **Find the derivative:** To find cr

1. **State the problem:** Find the critical numbers of the function $$p(t) = t e^{5t}$$. Critical numbers occur where the derivative is zero or undefined. 2. **Find the derivative:

1. **State the problem:** Find the absolute maximum and absolute minimum values of the function $$f(x) = x^{-2} \ln(x)$$ on the interval $$\left[\frac{1}{2}, 5\right]$$. 2. **Find

1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$$. 2. **Factor the numerator:** Recognize that $$x^2 - 1$$ is a difference of squares, so

1. The problem asks for the limit of the function $f(x) = \frac{1}{x}$ as $x$ approaches $0$ from the positive side, written as $\lim_{x \to 0^+} \frac{1}{x}$.\n\n2. When $x$ appro

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{2x + 1}{x + 3}$$. 2. **Analyze the expression:** As $$x$$ approaches infinity, the terms with the highest power

1. The problem asks for the limit of the function $$\frac{\sin x}{x}$$ as $$x$$ approaches 0. 2. This is a classic limit in calculus often used to define the derivative of the sine