91Ó°ÊÓ

Describe the interval(s) on which the function is continuous. $$ f(x)=x \sqrt{x+3} $$

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{3}-x^{2}+x-2, \quad[0,3], \quad f(c)=4 $$

If the functions \(f\) and \(g\) are continuous for all real \(x\), is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x\) ? If either is not continuous, give an example to verify your conclusion.

In the context of finding limits, discuss what is meant by two functions that agree at all but one point.

A dial-direct long distance call between two cities costs $$\$ 1.04$$ for the first 2 minutes and $$\$ 0.36$$ for each additional minute or fraction thereof. Use the greatest integer function to write the cost \(C\) of a call in terms of time \(t\) (in minutes). Sketch the graph of this function and discuss its continuity.

At 8:00 A.M. on Saturday a man begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 A.M. he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. [Hint: Let \(s(t)\) and \(r(t)\) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function \(f(t)=s(t)-r(t) .]\)

Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([1,5]\), there is one with a volume of 275 cubic centimeters.

Use the position function \(s(t)=-16 t^{2}+1000\), which gives the height (in feet) of an object that has fallen for \(t\) seconds from a height of 1000 feet. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). If a construction worker drops a wrench from a height of 1000 feet, how fast will the wrench be falling after 5 seconds?

Use the position function \(s(t)=-4.9 t^{2}+150\), which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\).