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Magazine Problem 48
Use a calculator to graph the function and estimate the value of the limit, then use L'H么pital's rule to find the limit directly. $$ \lim _{x \rightarrow 0^{+}} \tan \left(x^{x}\right) $$
Problem 49
For the following exercises, find the antiderivatives for the functions.\(\int \frac{x d x}{\sqrt{x^{2}+1}}\)
Problem 49
Use a calculator to graph the function and estimate the value of the limit, then use L'H么pital's rule to find the limit directly. $$ \lim _{x \rightarrow 0^{+}} \frac{\ln x}{\sin x} $$
Problem 50
In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. $$ y=\frac{x+1}{x^{2}+2 x+6} \text { over }[0,1] $$
Problem 50
In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{e^{t} \cos ^{-1}\left(e^{t}\right)}{\sqrt{1-e^{2 t}}} d t\)
Problem 50
Use a calculator to graph the function and estimate the value of the limit, then use L'H么pital's rule to find the limit directly. $$ \lim _{x \rightarrow 0} \frac{e^{x} e^{-x}}{x} $$
Problem 50
For the following exercises, find the antiderivatives for the functions.\(\int-\frac{d x}{x \sqrt{1-x^{2}}}\)
Problem 51
In the following exercises, compute each definite integral. \(\int_{0}^{1 / 2} \frac{\tan \left(\sin ^{-1} t\right)}{\sqrt{1-t^{2}}} d t\)
Problem 51
Find the area under \(y=1 / x\) and above the \(x\) -axis from \(x=1\) to \(x=4\) $$ \int \frac{d x}{x \ln (x) \ln (\ln x)}=\ln (\ln (\ln x))+C $$
Problem 52
In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. $$ y=-2^{-x} \text { over }[0,1] $$
