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Magazine Problem 1
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\sin x ; a=\frac{1}{6} \pi ; n=3 $$
Problem 1
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{0}^{+\infty} e^{-x} d x $$
Problem 1
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow+\infty} \frac{x^{2}}{e^{x}} $$
Problem 2
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\tan x ; a=0 ; n=3 $$
Problem 2
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{-\infty}^{1} e^{x} d x $$
Problem 3
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{-\infty}^{0} x 5^{-x^{a}} d x $$
Problem 3
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\sinh x ; a=0 ; n=4 $$
Problem 3
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 1 / 2^{-}} \frac{\ln (1-2 x)}{\tan \pi x} $$
Problem 4
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{1}^{+\infty} 2^{-x} d x $$
Problem 4
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\cosh x ; a=0 ; n=4 $$
