Functions

See: Trigonometric Functions Asymptotes Critical Points Critical Points (Multivariable) Taylor Series Gaussian Elimination Gauss-Jordan Elimination

Exercise

Consider the function $f(x) = x^2e^{-x^3}$. Mark true or false: a) The function is continuous b) The function is integrable in $[0, 1]$ c) The function is improperly integrable in $[1, +\infty)$ d) The function has a global maximum in $[0, +\infty)$
Consider the function $f(x) = \frac{x-1}{x^2+1}$. Mark true or false: a) The function is improperly integrable in $[2, +\infty)$ b) The function has a vertical asymptote c) The function has a local maximum and a local minimum d) The function has an oblique asymptote e) The function is increasing
Consider the function $f(x) = \frac{4-5x^2}{x^2+x-2}$. Mark true or false: a) The function is continuous in every $x \in \mathbb{R}$
Consider the function $f(x) = \frac{\sqrt{x}-x^2}{x}$. Mark true or false: a) $f$ is continuous in $\mathbb{R}$
Consider the function $f(x) = \frac{x^4-10x^2+9}{x^3}$. Mark true or false: a) $f$ is continuous in every $x \in \mathbb{R}$
Consider the function $f(x) = \frac{e^x}{e^x+3}$. Mark true or false: a) The function is continuous in every $x \in \mathbb{R}$ b) The function has a vertical asymptote c) The function is improperly integrable in $[1, +\infty)$ d) The function is increasing e) $F(x) = \ln(e^x+3)$ is an antiderivative of $f$ such that $F(0) = \ln 4$ f) $y = 0$ is a horizontal asymptote
Consider the function $f(x) = (x+1)e^{-x}$ Mark true or false: a) $f$ is well defined for every $x \in \mathbb{R}$ b) The function has a vertical asymptote c) The function has a global maximum in $[0, +\infty)$ d) The function has a horizontal asymptote e) $F(x) = 2 - e^{-x}(x+2)$ is an antiderivative of $f$ such that $F(0) = 2$ f) The function is improperly integrable in $[2, +\infty)$
Consider the function $f(x) = \frac{x+2}{x^2-1}$. Mark true or false: a) The function is improperly integrable in $[2, +\infty)$ b) The function has two vertical asymptotes c) The function has a local maximum and a local minimum d) The function has an oblique asymptote e) The function is decreasing for $x > 0$ f) $F(x) = \frac{3}{2}\ln(1-x) - \frac{1}{2}\ln(x+1) + 1$ is the antiderivative of $f$ such that $F(0) = 1$
Consider the function $f(x) = \frac{x^2-1}{x+2}$. Mark true or false: a) The function is continuous in $\mathbb{R}$ b) $x = 0$ is a horizontal asymptote c) The function is improperly integrable in $[1, +\infty)$ d) The function has a global maximum in $(0, +\infty)$ e) The function has a global minimum in $(-1, +\infty)$ f) $y = x-2$ is an oblique asymptote for $x \to +\infty$ g) $F(x) = \frac{1}{2}x^2 - 2x + 3\ln(x+2)$ is an antiderivative of $f$
Consider the function $f(x) = \frac{\sqrt{x+1}}{x-1}$. Mark true or false: a) The function is defined in ${x \in \mathbb{R} : x \neq 1}$ b) $y = x$ is an oblique asymptote c) The function is improperly integrable in $[2, +\infty)$ d) The function decreasing in the whole domain e) $x = 1$ is a vertical asymptote f) $f$ has a global maximum g) $F(x)=2\sqrt{x+1}+ \sqrt{2}\ln(\sqrt{x+1}-2)- \sqrt{2}\ln(\sqrt{x+1}+2)$ is an antiderivative of $f$
For each of the following functions find: a) Domain b) Asymptotes (if there are) c) Monotonicity intervals and convexity d) Classify the critical points e) Draw a qualitative graph
$f(x) = -2x^3-6x^2+5$
$f(x) = x^4-5x^2+4$
$f(x) = \sqrt{x}e^x$
$f(x) = x\sqrt{\frac{2x-1}{2x+1}}$
$f(x) = \frac{3x^2+x+3}{x^2+1}$
$f(x) = x^{2/3}(6-x)^{1/3}$
$f(x) = x \sqrt{9-x^2}$
$f(x) = \ln x - (\ln x)^2$
$f(x) = \left(5+\frac{1}{x^2}\right)^2 - \frac{8}{x^3}$
$f(x) = \sin^3(x)$
$f(x) = \ln x - 2x$
$f(x) = \ln( x - 1 ) + 1$ for $x \in [-2, 2]$
Given the following functions: a) $\ln|x|-2x$ b) $2x^3-9x^2+12x-1$ c) $x(x-1)^2$

i) Find the domain and asymptotes (if there are) ii) Detect the monotonicity intervals and classify the critical points iii) Collect all the obtained information on the cartesian plan drawing the graph