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Magazine Problem 69
Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ N(t)=130 \times 2^{1.2 t} $$
Problem 69
Which of the following functions is one to one (use the horizontal line test)? (a) \(f(x)=x^{2}, x \geq 0\) (b) \(f(x)=x^{2}, x \in \mathbf{R}\) (c) \(f(x)=\sqrt{x}, x \geq 0\) (d) \(f(x)=\ln x, x>0\) (e) \(f(x)=\frac{1}{x^{2}}, x \neq 0\) (f) \(f(x)=\frac{1}{x^{2}}, x>0\)
Problem 69
Show that the identity \(1+\tan ^{2} \theta=\sec ^{2} \theta\) follows from \(\sin ^{2} \theta+\cos ^{2} \theta=1\)
Problem 70
Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ I(u)=4.8 u^{-0.89} $$
Problem 70
(a) Show that \(f(x)=x^{3}-1, x \in \mathbf{R}\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\).
Problem 70
Show that the identity \(1+\cot ^{2} \theta=\csc ^{2} \theta\) follows from \(\sin ^{2} \theta+\cos ^{2} \theta=1\)
Problem 71
Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ R(t)=3.6 t^{1.2} $$
Problem 71
(a) Show that \(f(x)=x^{2}+1, x \geq 0\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\)
Problem 71
Solve \(\cos ^{2} \theta-2=2 \sin \theta\) on \([0,2 \pi)\).
Problem 72
Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ L(t)=2^{-2.7 y+1} $$
