∫ calculus

Problem statement: Find the minimum value of the function and the value of $\theta$ for which it occurs for $f(\theta)=(150\sin\theta+183.384)(123.5+150\cos\theta)$. 1. Write the f

1. Stating the problem. Find the minimum value of $f(\theta)=(150\sin\theta+183.384)(253.8-150\cos\theta)$ and find the value(s) of $\theta$ where this minimum occurs.

1. **Problem Statement:** Determine whether the function is continuous at the given number for Activity 1, part 1: $f(x) = 3x - 1$ at $x = 2$. 2. **Formula and Rules:** A function

1. **Problem Statement:** Determine whether the function $f(x) = 3x - 1$ is continuous at $x = 2$. 2. **Recall the definition of continuity at a point $x = c$:**

1. **State the problem:** We want to find $a$ and $\log_e L$ given the limit

1. **State the problem:** We need to find the derivative of the function $$f(x) = (4x^5 + 2x^3 - 5x)^5$$ using the chain rule. 2. **Recall the chain rule formula:** If $$f(x) = [g(

1. **State the problem:** We have a particle's position function given by $$f(t) = t^2 - \sin(2t)$$ for time $$t$$ in the interval $$0 \leq t \leq 3$$ seconds. We want to analyze t

1. **Problem:** Evaluate the third derivative $\frac{d^3}{dx^3} (4x^3 - 3x^2 + 2x + 10)$. 2. **Formula and rules:** The derivative of a polynomial term $ax^n$ is $a n x^{n-1}$. The

1. **Problem:** Evaluate the third derivative $\frac{d^3}{dx^3} (4x^3 - 3x^2 + 2x + 10)$. 2. **Formula and rules:** The derivative of $x^n$ is $nx^{n-1}$. The third derivative mean

1. **Problem Statement:** Find the differential coefficient of the function $$y = x^{\sin^{-1} x}$$ with respect to $$\sin^{-1} x$$. 2. **Understanding the problem:** We want to fi

1. **Problem Statement:** Find the differential coefficient of the function $$y = x^{\sin^{-1} x}$$ with respect to $$\sin^{-1} x$$. 2. **Understanding the problem:** We want to fi

1. **Problem:** Calculate the limit $$\lim_{x \to -1} \frac{\sqrt{x^3 - 2}}{x - 1}$$. 2. **Formula and rules:** To find limits involving square roots and rational expressions, chec

1. **State the problem:** Find the value(s) of $c$ in the interval $[0,2]$ such that the derivative $f'(c) = 0$ for the function $f(x) = x^2 + 2x$. 2. **Find the derivative:** The

1. Problem 4.1: Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$. 2. Use the product rule for derivatives: if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)

1. **Problem 4.1:** Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$. 2. **Formula and rules:** Use the product rule for derivatives:

1. Find the derivative of the function $f(x) = (2x - 1)^2 \sin(2x)$.\n\nStep 1: State the problem: Differentiate $f(x) = (2x - 1)^2 \sin(2x)$.\nStep 2: Use the product rule: If $f(

1. **Problem a:** Find $$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$$ 2. **Step 1:** Recognize that direct substitution $$x=4$$ gives $$\frac{16 - 16}{4 - 4} = \frac{0}{0}$$ which is in

1. **Problem:** Given the derivative $f'(x) = x^2 (x - 2)(x - 3)^2 (x - 4)$, find an open interval where $f(x)$ is decreasing. 2. **Recall:** A function $f(x)$ is decreasing where

1. **State the problem:** Find the area of the region bounded by the graph of the function $f(x) = x^3 + 1$, the x-axis, and the vertical lines $x = -2$ and $x = 0$. 2. **Formula a

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt$$. 2. This integral involves a composition of trigonometric functions and a polynomial term, which

1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt.$$\n\n2. This integral is quite complex because it involves a composition of the tangent function i