4.7: Summary of Chapter 4

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We have defined continuity of a function, shown that if a function \(F\) has a derivative at a point \(x\) in its domain then \(F\) is continuous at \(x\), and used this property to prove the Power Chain Rule,

\[\text { PCR: } \quad\left[u^{n}(t)\right]^{\prime}=n u^{n-1}(t) \times u^{\prime}(t)\]

We proved PCR for all positive integers, \(n\). In Exercises 4.3.3 and 4.3.4 you showed PCR to be true for all rational numbers, \(n\). In fact, PCR is true for all numbers, \(n\). We then used the power chain rule to solve some problems.

Exercises for Chapter 4, Continuity and the Power Chain Rule.

Chapter Exercise 4.7.1 Compute the derivative of \(P\).

\(P(t) = 3t ^{2} - 2t + 7\)
\(P(t) = t + \frac{2}{t}\)
\(P(t) = \sqrt{t + 2}\)
\(P(t) = (t ^{2} + 1 )^5\)
\(P(t) = \sqrt{2t + 1}\)
\(P(t) = 2t ^{-3} - 3t ^{-2}\)
\(P(t) = \frac{5}{t + 5}\)
\(P(t) = (1 + \sqrt{t})^{-1}\)
\(P(t) = ( 1 + 2t) ^5\)
\(P(t) = ( 1 + 3t) ^{1/3}\)
\(P(t) = \frac{1}{1 + \sqrt{t}}\)
\(P(t) = \sqrt[3] {1 + 2t}\)

Chapter Exercise 4.7.2 In "Natural History", March, 1996, Neil de Grass Tyson discusses the discovery of an astronomical object called a "brown dwarf".

"We have suspected all along that brown dwarfs were out there. One reason for our confidence is the fundamental theorem of mathematics that allows you to declare that if you were once 3'8" tall and are now 5'8" tall, then there was a moment when you were 4'8" tall (or any other height in between). An extension of this notion to the physical universe allows us to suggest that if round things come in low-mass versions (such as planets) and high-mass versions (such as stars) then there ought to be orbs at all masses in between provided a similar physical mechanism made both.

What fundamental theorem of mathematics is being referenced in the article about the astronomical objects called brown dwarfs? What implicit assumption is being made about the sizes of astronomical objects? (For future consideration: Is the number of 'orbs' countable?)

Chapter Exercise 4.7.3 In a square field with sides of length 1000 feet that are already fenced a farmer wants to fence two rectangular pens of equal area using 400 feet of new fence and the existing fence around the field. What dimensions of lots will maximize the area of the two pens?

Chapter Exercise 4.7.4 You must cross a river that is 50 meters wide and reach a point on the opposite bank that is 1 km up stream. You can travel 6 km per hour along the river bank and 1 km per hour in the river. Describe a path that will minimize the amount of time required for your trip. Neglect the flow of water in the river.

Chapter Exercise 4.7.5 Find the point of intersection of the tangents to the ellipse \(x ^{2}/224 + y ^{2}/128 = 1/7\) at the points (2, 4) and (5, -2).