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Chapter 5: Problem 6
Evaluate the expression without using a calculator. \(\arcsin 0\)
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Let \(f\) and \(g\) be one-to-one functions. Prove that (a) \(f \circ g\) is one-to- one and (b) \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\).
Find the indefinite integral using the formulas of Theorem \(5.20 .\) $$ \int \frac{1}{1-4 x-2 x^{2}} d x $$
Solve the differential equation. $$ \frac{d y}{d x}=\frac{1}{(x-1) \sqrt{-4 x^{2}+8 x-1}} $$
Prove that if a function has an inverse function, then the inverse function is unique.
Graph \(y_{1}=\frac{x}{1+x^{2}}, y_{2}=\arctan x\), and \(y_{3}=x\) on \([0,10]\).
Prove that \(\frac{x}{1+x^{2}}<\arctan x
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