∫ calculus

1. **State the problem:** We need to find the area between the curves given by the integral $$\int_{-1}^{1} [(-2x^2 + 8x + 11) - (8x + 9)] \, dx$$. 2. **Simplify the integrand:** S

1. **Problem Statement:** Estimate the limit \(\lim_{t \to 0} \frac{e^t - 1}{t}\) using a table of values and graph. 2. **Formula and Important Rule:**

1. **Δίνεται το πρόβλημα:** Να δείξουμε ότι η παράγωγος της συνάρτησης $f(x) = e^x + \frac{x^2}{2} - x - 1$ είναι αντιστρέψιμη και να βρούμε το πεδίο ορισμού της αντίστροφης $(f')^

1. **Problem statement:** Evaluate the integral $$\int \frac{dx}{5 + f\left(\frac{\pi}{2} - x\right)}$$ where $f(x) = \tan x$. 2. **Substitute the function:** Since $f(x) = \tan x$

1. The problem states that $f$ is continuous on the interval $[1,5]$ with $f(1) = 5$ and $f(5) = 2$. 2. The Intermediate Value Theorem (IVT) says that if a function is continuous o

1. Consider the function $h(x) = e^{-g(x)}$ where $g(x)$ is continuous with a continuous first derivative and has a local maximum at $x=a$. 2. Since $g(x)$ has a local maximum at $

1. The problem is to find the integral of the function $d(e^{g(x)})dx$. 2. This expression suggests the differential of $e^{g(x)}$ with respect to $x$, which is $d(e^{g(x)}) = e^{g

1. The problem is to understand the expression $e^{g(x)} \, dx$ and what it represents. 2. Here, $e^{g(x)}$ means the exponential function with the exponent being the function $g(x

1. **Problem Statement:** Find the extreme values (absolute and local) of the function $$y = \frac{x^2}{x+2}$$ over its natural domain. 2. **Domain:** The function is undefined whe

1. **Problem statement:** Consider the function $h(x) = e^{-g(x)}$ where $g(x)$ is continuous with a continuous first derivative. If $g(x)$ has a local maximum at the point $a$, wh

1. **State the problem:** Find the differential of the function $$y=\frac{3x - 7}{5x + 4}$$. 2. **Recall the formula:** For a function $$y=\frac{u}{v}$$, the derivative is given by

1. **State the problem:** Find the first-order derivative of the function $$f(x) = (x^2 + 3)(x^2 - 2)$$. 2. **Recall the product rule:** For two functions $$u(x)$$ and $$v(x)$$, th

1. **State the problem:** We are given the implicit equation $$\psi_5(x,y) = x \cdot (xy) - 1 = 0$$ and asked to find the derivative $\frac{dy}{dx}$. 2. **Rewrite the equation:** S

1. The problem gives us the limits and function value at $x=2$ for the function $f(x)$ with domain $-2 \leq x < 4$. 2. We are asked to analyze the behavior of $f(x)$ near $x=2$ and

1. The problem asks to find the limit as $x$ approaches $-1$ of the function $$f(x) = \frac{x^2 - 1}{x + 1}.$$\n\n2. First, note that directly substituting $x = -1$ into $f(x)$ giv

1. **State the problem:** Differentiate implicitly the equation $$x^2 + y^2 = \log(xy)$$ with respect to $x$. 2. **Recall the rules:**

1. **Problem:** Find the derivative $\frac{dy}{dx}$ for $y = \log_4 (3x - 4)$. 2. **Formula and rules:** The derivative of $y = \log_a u$ is given by

1. **State the problem:** We analyze the sales tax function $T(x)$ at specific years $x$ to find limits and values based on the graph description. 2. **Recall limit definitions:**

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{x^2}{1 - 8x^2}$$. 2. **Recall the rule for limits at infinity for rational functions:** When $x$ approaches in

1. The problem asks to find the limit of the function $T(x)$ as $x$ approaches 60, i.e., $\lim_{x \to 60} T(x)$. 2. From the graph description, $T(x)$ is a stepwise function repres

1. **Problem:** Differentiate $y = \frac{3}{\sqrt{8x+3}}$ with respect to $x$. 2. **Formula and rules:** Recall that $\sqrt{u} = u^{1/2}$ and the derivative of $u^n$ with respect t