91Ó°ÊÓ

Suppose that the daily cost \(C\) of manufacturing bicycles is given by \(C(x)=80 x+5000 .\) Then the average daily cost \(\bar{C}\) is given by \(\bar{C}(x)=\frac{80 x+5000}{x} .\) How many bicycles must be produced each day for the average cost to be no more than \(\$ 100 ?\)

Find bounds on the real zeros of each polynomial function. $$ f(x)=x^{4}+x^{3}-x-1 $$

Challenge Problem Gravitational Force According to Newton's Law of Universal Gravitation, the attractive force \(F\) between two bodies is given by $$F=G \frac{m_{1} m_{2}}{r^{2}}$$ where \(m_{1}, m_{2}=\) the masses of the two bodies \(r=\) distance between the two bodies \(G=\) gravitational constant \(=6.6742 \times 10^{-11}\) newtons " meter \(^{2}\). kilogram \(^{-2}\) Suppose an object is traveling directly from Earth to the moon. The mass of Earth is \(5.9742 \times 10^{24}\) kilograms, the mass of the moon is \(7.349 \times 10^{22}\) kilograms, and the mean distance from Earth to the moon is 384,400 kilometers. For an object between Earth and the moon, how far from Earth is the force on the object due to the moon greater than the force on the object due to Earth?

Find bounds on the real zeros of each polynomial function. $$ f(x)=3 x^{4}-3 x^{3}-5 x^{2}+27 x-36 $$

Create a rational function with the following characteristics: three real zeros, one of multiplicity \(2 ; y\) -intercept 1 ; vertical asymptotes, \(x=-2\) and \(x=3 ;\) oblique asymptote, \(y=2 x+1\). Is this rational function unique? Compare your function with those of other students. What will be the same as everyone else's? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the rational function?

Explain the circumstances under which the graph of a rational function has a hole.

Find bounds on the real zeros of each polynomial function. $$ f(x)=4 x^{5}+x^{4}+x^{3}+x^{2}-2 x-2 $$

Find bounds on the real zeros of each polynomial function. $$ f(x)=-x^{4}+3 x^{3}-4 x^{2}-2 x+9 $$

Use the Intermediate Value Theorem to show that each polynomial function has a real zero in the given interval. $$ f(x)=2 x^{3}+6 x^{2}-8 x+2 ;[-5,-4] $$

Use the Intermediate Value Theorem to show that each polynomial function has a real zero in the given interval. $$ f(x)=x^{5}-x^{4}+7 x^{3}-7 x^{2}-18 x+18 ;[1.4,1.5] $$