∫ calculus

1. The problem is to understand and apply the derivative method in calculus. 2. The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

1. **Problem 1:** Evaluate $$\int_{-3}^0 -8x (2x^2 + 3)^2 \, dx$$ with substitution $$u = 2x^2 + 3$$. 2. **Formula and substitution:**

1. **State the problem:**\nWe have two functions:\n$$g(x) = (x+1)^2(x-2)^2$$\nand\n$$h(x) = (x+1)(x-1)(x+2)(x-2)$$\nWe want to find transformations of $h$ so that:\n\ni. Using tran

1. **State the problem:** We need to estimate the difference quotient $$\frac{f(2+h)-f(2)}{h}$$ for $h=1$. This expression represents the average rate of change of the function $f(

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{\sqrt{x} - 5}{\sqrt{x} + 5}$$. 2. **Recall the formula:** For a function $$f(x) = \frac{u(x)

1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = (2 - x - 3x^3)(7 + x^5)$$. 2. **Formula used:** To differentiate a product of two function

1. **State the problem:** We need to find the rate of change of the average rates of change for each function over consecutive equal-length input intervals. 2. **Recall the concept

1. **Problem Statement:** Sketch the graph of a function $f$ satisfying the conditions for problem 5:

1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches infinity, negative infinity, and the points $x=1$ and $x=3$. We also need to find

1. **State the problem:** We want to find the limit as $x \to \infty$ of $$\left(\frac{4^x + 9^x}{x}\right)^{\frac{1}{x}}.$$\n\n2. **Analyze the expression inside the parentheses:*

1. **State the problem.** We want to find

1. **State the problem:** Find the limit $$\lim_{x \to \infty} \left( \frac{4^x + 9^x}{x} \right)^{\frac{1}{x}}.$$

1. **Problem Statement:** Find the limits and continuity properties as requested in the exercises. ---

1. **Problem statement:** Find the limits and continuity values as requested in Exercise 1.1 no.4. 2. **(a) Find** $\lim_{x \to 2^-} f(x)$: This is the left-hand limit as $x$ appro

1. **State the problem:** We need to find the derivative of the function $f(x) = 2 \sin x + \sin^2 x$ at $x=0$. 2. **Recall the derivative rules:**

1. **State the problem:** We want to calculate the indefinite integral $$\int \frac{7\cos(x)\sin^2(2x)}{8} \, dx.$$ 2. **Apply the half-angle formula:** Recall that $$\sin(2x) = 2\

1. **State the problem:** We want to calculate the indefinite integral $$\int \frac{7\cos(x)\sin(4x^2)}{8} \, dx.$$ 2. **Analyze the integrand:** The integral contains $\cos(x)$ an

1. **State the problem:** We need to evaluate the integral $$\int \frac{\arctan(\sqrt{x})}{\sqrt{x}} \, dx$$. 2. **Substitution:** Let $$t = \sqrt{x} = x^{1/2}$$, so $$x = t^2$$ an

1. **State the problem:** We need to evaluate the definite integral $$\int_0^{5\pi} x^3 \sin(x) \, dx$$. 2. **Recall the formula:** To integrate products of powers of $$x$$ and tri

1. **State the problem:** We need to determine the open intervals where the function is increasing, decreasing, or constant based on the given graph. 2. **Recall definitions:**

1. **State the problem:** We need to find $\delta$ such that whenever $0 < |x - 3| < \delta$, it follows that $|f(x) - 2| < 0.5$. 2. **Understand the inequality:** The condition $|