91Ó°ÊÓ

Let $$ f(x)=4 \sin (2 \pi x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

If \(z=a+b i\), find \(z+\bar{z}\) and \(z-\bar{z}\).

Let $$ f(x)=-\frac{3}{2} \sin \left(\frac{\pi}{3} x\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

If \(z=a+b i\), find \(\bar{z}\). Use your answer to compute \(\overline{(\bar{z})}\), and compare your answer with \(z\).

Logistic Transformation Suppose that $$ f(x)=\frac{1}{1+e^{-(b+m x)}} $$ A function of the form (1.8) is called a logistic function. The logistic function was introduced by the Dutch mathematical biologist Verhulst around 1840 to describe the growth of populations with limited food resources. Show that $$ \ln \frac{f(x)}{1-f(x)}=b+m x $$ This transformation is called the logistic transformation. It is a standard transformation for linearizing functions of the form (1.8).

Not every study of species richness as a function of productivity produces a hump-shaped curve. Owen (1988) studied rodent assemblages in Texas and found that the number of species was a decreasing function of productivity. Sketch a graph that would describe this situation.

Let $$ f(x)=4 \cos \left(\frac{x}{4}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Solve each quadratic equation in the complex number system. \(2 x^{2}-3 x+2=0\)

Solve each quadratic equation in the complex number system. \(3 x^{2}-2 x+1=0\)

Let $$ f(x)=7 \cos (2 x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).