∫ calculus

1. **State the problem:** Given the function $$f(x) = \frac{5}{4\sqrt{x}} - \frac{\sqrt{x^3}}{4}$$, find the derivative $$f'(x)$$ and then evaluate $$f'(1)$$. Express the answer as

1. Evaluate $$\lim_{x \to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$$. Step 1: Recognize this limit is of the form $$\frac{f(x) - f(2)}{x - 2}$$ where $$f(x) = \frac{1}{x}$$.

1. **State the problem:** We are given the function $$f(x) = \frac{5}{4\sqrt{x}} - \frac{3\sqrt{x^3}}{4}$$ and need to find its derivative at $$x=1$$, i.e., $$f'(1)$$. The answer s

1. **State the problem:** We are given the function $$f(x) = \frac{1}{\sqrt{x}} + \frac{3\sqrt{x}}{2}$$ and need to find its derivative at $$x=6$$, i.e., $$f'(6)$$, expressed as a

1. **State the problem:** We are given the function $$f(x) = - \frac{4}{3\sqrt{x}} - 2\sqrt{x}$$ and need to find its derivative at $$x=2$$, expressed as a single fraction in simpl

1. Problem: Find the limit $$\lim_{x \to -7} (2x + 5)$$ Step 1: Substitute $x = -7$ directly since the function is linear and continuous.

1. The problem asks us to find the derivative of the function $$f(x) = -\frac{1}{2x^3}$$ and express the answer using negative exponents. 2. First, rewrite the function using negat

1. The problem is to find the derivative of the function $$f(x) = -\frac{4\sqrt{x}}{3}$$. 2. First, rewrite the square root in exponent form: $$\sqrt{x} = x^{\frac{1}{2}}$$.

1. **State the problem:** We are given the function $f(x) = \frac{4}{5x^5}$ and need to find its derivative $f'(x)$. 2. **Rewrite the function:** Express $f(x)$ with negative expon

1. Evaluate each limit using limit theorems: (i) $$\lim_{x \to 3} (2x + 4) = 2(3) + 4 = 6 + 4 = 10$$

1. **State the problem:** We want to show that the functions defined by the series

1. The user provided several limit expressions and functions to analyze. 2. Let's clarify and solve each limit step-by-step.

1. The problem is to find the indefinite integral of the polynomial function $$3x^2 + 7x - 2$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n \, dx = \

1. We are given two functions \(f(x)\) and \(g(x)\). \(f(x)\) is approximately linear through points (-4, -1.5), (0,0), (3,2), (5,3) and \(g(x)\) goes through (-4,4), (-2,-2), (0,-

1. **Problem:** (a) Show that $$\cos 2A = 1 - 2 \sin^2 A$$ using a formula from page 2.

1. We are asked to find the derivative of the function $$f(x) = (x-3)^3 (x+1)$$. 2. This is a product of two functions: $$u = (x-3)^3$$ and $$v = (x+1)$$. We will use the product r

1. **State the problem:** Differentiate the curve given by $$y = \frac{(3x^2 - 5)^{\frac{1}{3}}}{x+4}$$

1. Problem: Evaluate the integral $$\int \frac{\cos 2x}{\sqrt{\sin 2x + 2}}\,dx$$ Step 1: Use substitution. Let $$u = \sin 2x + 2$$.

1. **State the problem:** We want to estimate the limit of the function $g(x)$ as $x$ approaches 3, i.e., compute $\lim_{x \to 3} g(x)$. 2. **Look at the graph near $x=3$ from the

1. The problem is to find the limit: $$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ for $x \neq -1$.

1. Problem: Find the limit $$\lim_{x \to -1} 3$$. 2. Explanation: The function here is the constant function $$f(x) = 3$$.