∫ calculus

1. **State the problem:** Find the degree 3 Taylor polynomial $T_3(x)$ of the function $$f(x) = (-7x + 109)^{\frac{5}{4}}$$ at the point $a=4$. 2. **Recall the Taylor polynomial fo

1. **State the problem:** We want to find the Taylor series expansion of the function $$f(x) = x^3$$ about the point $$x=2$$ and determine the first five coefficients $$c_0, c_1, c

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \frac{1 - x^2}{x^2 + 3}$$. 2. **Recall the rule for limits at infinity:** When $x$ approaches $\pm \infty$, the high

1. Das Problem lautet: Bestimmen Sie das Integral der Funktion $p(x) = 2e^{-0,5x}$ von $x = -1$ bis $x = 3$. 2. Die Formel für das Integral einer Exponentialfunktion $e^{ax}$ ist $

1. **Problem (i):** Given $y = \frac{x^2 - x}{e^x}$, find $\frac{dy}{dx}$ and simplify. 2. **Formula:** Use the quotient rule for derivatives: $$\frac{d}{dx}\left(\frac{u}{v}\right

1. **Stating the problem:** We want to evaluate the integral $$\int \frac{-2}{2x^2 - x^3 - x} \, dx$$. 2. **Simplify the denominator:** Factor the denominator:

1. **Problem:** Calculate the definite integral $$\int_{-2}^1 \frac{4}{x^2} \, dx$$. 2. **Formula and rules:** The integral of $$\frac{1}{x^2}$$ is $$\int x^{-2} dx = -x^{-1} + C =

1. **State the problem:** We need to evaluate the integral $$\int x e^{-2x} \, dx$$. 2. **Formula and method:** We will use integration by parts, which states:

1. **State the problem:** Find the limit as $x$ approaches $+\infty$ of the expression $$\lim_{x \to +\infty} \frac{3 - x}{\sqrt[3]{8x^3 - 3x^2 + x - 4}}.$$\n\n2. **Recall the form

1. **Problem statement:** Evaluate the integral $$\int_{-1}^1 f(x) \, dx$$ where $$f(x) = e^{\sqrt[3]{x}} - 4x$$, given that $$\int_0^1 f(x) \, dx = 3e - 8$$ and $$\int_0^{-1} f(x)

1. The problem is to evaluate the definite integral $$\int_1^7 x \, dx$$. 2. The formula for the integral of $$x$$ is $$\int x \, dx = \frac{x^2}{2} + C$$, where $$C$$ is the const

1. **State the problem:** Differentiate the function $$y = 7x^5 - 4x^3 + 9x - 2$$ with respect to $$x$$. 2. **Recall the power rule for differentiation:** If $$y = x^n$$, then $$\f

1. **State the problem:** Given the temperature function during illness $$T(t) = -0.6t^2 + 0.67t + 37$$ where $T$ is temperature in degrees Celsius and $t$ is time in days, we need

1. **State the problem:** Find the integral of $5x\cos x\,dx$. 2. **Formula and method:** Use integration by parts, which states:

1. The problem is to create a cheat sheet for derivatives to help with a math test. 2. The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x

1. **State the problem:** We want to find the limit as $h \to 0$ of the difference quotient for $f(x) = 4x + x^2$ at $x=2$, which is $$\lim_{h \to 0} \frac{f(2+h) - f(2)}{h}.$$ Als

1. **State the problem:** Find the slope of the tangent line at $x=2$ for the functions $f(x) = 4x - x^2$ and $g(x) = \frac{1}{3x - 7}$ using the definition of the derivative. 2. *

1. The problem is to simplify or integrate an expression using the substitution $u=9-x^2$. 2. The substitution method involves replacing a complicated expression with a simpler var

1. **State the problem:** We need to evaluate the integral $$\int \frac{x^3}{\sqrt{9 - x^2}} \, dx$$. 2. **Recall the formula and substitution:** When dealing with integrals involv

1. **State the problem:** We need to evaluate the integral $$\int e^{2x} \cos x \left(e^{2x}\right) \, dx$$. 2. **Simplify the integrand:** Notice that the integrand is $$e^{2x} \c

1. **State the problem:** We need to find the integral of the function $e^{2x}$ with respect to $x$. 2. **Formula and rules:** The integral of an exponential function $e^{ax}$ is g