∫ calculus

1. **State the problem:** We are given the difference quotient expression for a function $f$: $$f(x + h) - f(x) = -2hx^2 + 5hx + 7h^2x + 5h^2 + 3h^3$$

1. **State the problem:** We are given a piecewise function $f(x)$ and need to find for each $a = -5, -1, 2, 5$ the values of $f(a)$ and the limit $\lim_{x \to a} f(x)$ if they exi

1. The problem is to find the limit: $$\lim_{x \to 0} \frac{\ln(1 + 3x)}{3 \sin(x)}$$ 2. To solve this, we use the property that we can multiply and divide by the same expression t

1. **State the problem:** We are given the function $$f(x) = \frac{1}{2x^2} + x^2 - \sqrt{x} + 5$$ for $$x > 0$$ and need to find its derivative $$f'(x)$$. 2. **Recall derivative r

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x^3}{\sqrt{x}}$$ and express it in the form $$f'(x) = ax^n$$, then find the exact value of t

1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt[3]{x^4}$$ and express it in the form $$f'(x) = ax^n$$, then find the exact value of the expo

1. **State the problem:** Find the derivative of the function $$f(x) = x^5 - 3\sqrt{x} + \frac{1}{x^2}$$. 2. **Rewrite the function using exponents:**

1. **Stating the problem:** We want to find the limit $$\lim_{x \to 0} \frac{\ln(1 + 3x)}{3 \sin(x)}$$

1. The problem asks to find the derivative of the function $f(x) = \sqrt{x}$ with respect to $x$. 2. Recall that $\sqrt{x} = x^{\frac{1}{2}}$.

1. **State the problem:** We want to find the integral of the function $$ (3x^2 + 2x) \sin(x^3 + x^2) \, dx $$. 2. **Identify the method:** This integral suggests using substitutio

1. **State the problem:** We want to estimate the limit of the function $$f(x) = \frac{2x}{x - 1}$$ as $$x$$ approaches infinity, i.e., $$\lim_{x \to \infty} \frac{2x}{x - 1}$$. 2.

1. **State the problem:** Evaluate the limit $$\lim_{h \to 1} \frac{h^2 - 1}{h - 1}$$. 2. **Recall the formula and rules:** When direct substitution leads to an indeterminate form

1. **Stating the problem:** Evaluate the integral $$\int (2x + 1) \cos(x^2 + x) \, dx$$. 2. **Formula and substitution rule:** When integrating a function of the form $$f(g(x)) \cd

1. **State the problem:** We need to find the indefinite integral $$\int \frac{x}{\sqrt{x^2 + 1}} \, dx$$. 2. **Recall the formula and rules:** A useful substitution for integrals

1. **State the problem:** We need to find the absolute minimum and maximum values of the function $$f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6$$ on the closed interval $$[-0.5, 1.2

1. **State the problem:** We need to find the absolute minimum and maximum values of the function $$f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6$$ on the closed interval $$[-0.5, 1.2

1. **State the problem:** We need to find the limit $$\lim_{x \to 0} \frac{1 - \cos x}{x^2}$$

1. **Problem Statement:** Find $r'(t)$, the velocity function of the rocket.

1. **Problem:** Find the first and second derivatives of the function $f(x) = x^6$. 2. **Formula:** The derivative of $x^n$ with respect to $x$ is given by the power rule:

1. The problem is to understand where a function decreases on its graph. 2. A function decreases on intervals where its derivative is less than zero, i.e., $f'(x) < 0$.

1. **State the problem:** We need to find the integral $$\int \frac{x^3 + 5}{x^2 - 25} \, dx$$. 2. **Rewrite the denominator:** Note that $$x^2 - 25 = (x - 5)(x + 5)$$.