91Ó°ÊÓ

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1} \frac{1-x+\ln x}{1+\cos \pi x}$$

Differentiate the function. $$ y=\tan [\ln (a x+b)] $$

Find the derivative of the function. Simplify where possible. \(y=\left(\tan ^{-1} x\right)^{2}\)

If \(\cosh x=\frac{5}{3}\) and \(x>0,\) find the values of the other hyperbolic functions at \(x\).

Find the derivative of the function. Simplify where possible. $$g(x)=\sqrt{x^{2}-1} \sec ^{-1} x$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$$\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(4 x)}$$

(a) If \(\$ 1000\) is borrowed at 8\(\%\) interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose \(\$ 1000\) is borrowed and the interest is compounded continuously. If \(A(t)\) is the amount due after \(t\) years, where \(0 \leqslant t \leqslant 3,\) graph \(A(t)\) for each of the interest rates \(6 \%, 8 \%,\) and 10\(\%\) on a common screen.

Differentiate the function. $$ H(z)=\ln \sqrt{\frac{a^{2}-z^{2}}{a^{2}+z^{2}}} $$

If \(\$ 3000\) is invested at 5\(\%\) interest, find the value of the investment at the end of 5 years if the interest is compounded (a) annually (b) semiannually (c) monthly (d) weekly (e) daily (f) continuously

Find the derivative of the function. Simplify where possible. \(y=\sin ^{-1}(2 x+1)\)