∫ calculus

1. **State the problem:** Given parametric equations $x = te^{2t}$ and $y = t^2 + t + 3$, we need to (a) show that $\frac{dy}{dx} = e^{-2t}$ and (b) show that the normal to the cur

1. **State the problem:** Given the curve defined by the equation $$x^3 + 3x^2y - y^3 = 3,$$ we need to:

1. **State the problem:** We need to find the $x$-coordinate of the stationary point of the curve given by $$y = \cos x \sin 2x$$ in the interval $$0 < x < \frac{1}{2} \pi$$, corre

1. **State the problem:** Find the $x$-coordinates of the stationary points of the curve given by $$y = e^{-5x} \tan^2 x$$ for $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$. 2. **Find the

1. **State the problem:** Differentiate the function $$f(x) = (2x^2 + 3)((x^5 - x + 2)^3)$$ with respect to $$x$$. 2. **Identify the rule:** This is a product of two functions, so

1. **State the problem:** Differentiate the function $$f(x) = \frac{x}{1-x} - \frac{3x^3}{4}$$ with respect to $$x$$. 2. **Differentiate the first term:** Use the quotient rule for

1. **State the problem:** Differentiate the function $$f(x) = \frac{\sqrt{3+x}}{\sqrt[3]{x^2-2}}$$ with respect to $$x$$. 2. **Rewrite the function using exponents:**

1. **Stating the problem:** We want to find the integral $$\int x^2 e^x \, dx$$. 2. **Method:** Use integration by parts, where we let:

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