∫ calculus

1. The problem is to show the derivative of a function $f(x)$ using first principles, which means using the definition of the derivative as a limit. 2. By definition, the derivativ

1. **Stating the problem:** Evaluate the limit $$\lim_{x\to \infty} \left(1 + \frac{5}{3x}\right)^{\frac{x}{2}}.$$\n\n2. **Rewrite the expression:** Notice the limit has the form s

1. We are asked to evaluate the limit $\lim_{x \to \infty} \left(1 + \frac{5}{3x}\right)$.\n\n2. As $x$ approaches infinity, the term $\frac{5}{3x}$ approaches $0$ because the deno

1. Problem a: Find $$\lim_{n \to \infty} \sqrt[n]{5n + 3}.$$ We rewrite as $$ (5n+3)^{1/n} = e^{\frac{1}{n} \ln(5n+3)}.$$

1. Stating the problem: Evaluate the limit $\lim_{n \to \infty} n \sqrt{5n + 3}$. 2. Rewrite the expression inside the limit:

1. We are asked to find the derivative $f'(x)$ of the function $f(x) = 4x + 7$ using the definition of the derivative. 2. The definition of the derivative is:

1. We need to find the derivative $f'(x)$ of the function $f(x) = 4x + 7$ using the definition of the derivative. 2. The definition of the derivative at a point $x$ is given by:

1. **State the problem:** We want to find the limit $$\lim_{x \to \infty} \frac{5^{x+1} + 7^{x+1}}{5^x - 7^x}$$ without using differentiation or L'Hopital's Rule.

1. **State the problem:** We are given three functions: $$f_1(x) = 2x^{6} \sin(x), \quad f_2(x) = x^{5} \cdot 5^{x}, \quad f_3(x) = x^{5} \ln(x) \cos(x)$$

1. **State the problem:** We have the parabola given by $$y = (9 - x)(x - 3)$$ and the horizontal line $$y = k - 3$$ where $$k > 3$$. We need to find the area of the shaded region

1. **Problem Statement:** (i) Find the integral $\int \sqrt{4 + x} \, dx$ without a calculator.

1. State the problem: Find the limit $$\lim_{x \to 7} (2x + 2)$$. 2. Since the function $2x + 2$ is a polynomial (which is continuous everywhere), the limit as $x$ approaches 7 is

1. The problem gives a differential equation: $$\frac{d w}{d x} = -kW^{3} T^{-\frac{1}{2}}.$$\n\n2. We want to solve this differential equation for $w$ as a function of $x$. Howeve

1. Stated Problem: Differentiate the function $$H(x) = 3 \sec x (1 - \tan x)$$. 2. Use the product rule for differentiation: If $$H(x) = f(x)g(x)$$, then $$H'(x) = f'(x)g(x) + f(x)

1. Stating the problem: We are asked to find two limits based on the description of the graph of a function $f(x)$. 2. For part (a), find $\lim_{x \to 0} f(x)$.

1. **State the problem:** Find the limit $$\lim_{x\to 1} \frac{\sqrt{x+3} - \sqrt{2x-1}}{x - 1}$$

1. **Stating the problem:** We want to find the limit $\lim_{z \to n\pi} \frac{z - n\pi}{\sin z}$. 2. **Substitution check:** Direct substitution yields $\frac{n\pi - n\pi}{\sin(n\

1. Najprv si uvedomme vzorec pre Taylorovu radu funkcie $\sin x$ okolo bodu $z$: $$\sin x = \sin z + \cos z (x-z) - \frac{\sin z}{2!} (x-z)^2 - \frac{\cos z}{3!} (x-z)^3 + \cdots$$

1. Stating the problem: We analyze multiple limit problems based on given piecewise and plotted graphs to find values of $a$ or certain limits and function values. 2. For problem 1

1. Evaluate $$ \int \frac{x^3}{\sqrt{x^2 + 1}} \, dx $$. 2. Evaluate $$ \int \sin^2{(x)} \cos^2{(x)} \, dx $$.

1. **Problem (a):** Evaluate $$\int \frac{x^3}{\sqrt{x^2 + 1}} \, dx$$ Step 1: Use substitution. Let $$u = x^2 + 1$$, so $$du = 2x \, dx$$ which gives $$x \, dx = \frac{du}{2}$$.