Homework Homework Assignment #28 Read Section 4.7 Page 265, Exercises: 31, 39, 43, 47, 53, 59 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 31.Let r be the monthly rent in an apartment building with 100 units. All units can be rent when r = $900 and one unit becomes vacant for every $10 increase in rent. The average monthly maintenance cost per occupied unit is $100. (a) Show that the number of units rented is n = 190 – r/10 for 900 ≤ r ≤ (b) Find a formula for the net cash intake (revenue minus maintenance) and determine the rent that maximizes intake.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A five-sided box is to be constructed from a rectangle of cardboard of sides A and B by cutting squares of side h from each corner and folding up the resulting tabs. 39.Find the value of h that maximizes the volume of the box if A = 15 and B = 24. What are the dimensions of the resulting box?

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 43.Use calculus to show that among all right triangles with hypotenuse of length 1, the isosceles triangle has maximum area.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 43.Can you see more directly why this must be true from Figure 23? If the diameter of the semicircle is 1, and the vertex is any point on the circumference, the angle at the vertex is 90º. When x = y, h has the largest value and so does area.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 47.Find the area of the largest isosceles triangle that can be inscribed in a circle of radius r.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 53.A small blood vessel of radius r branches off at an angle of θ from a larger vessel of radius R to supply blood along a path from A to B. According to Poiseuille’s Law, the total resistance to blood flow is proportional to where a and b are as in Figure 26. Show that total resistance is minimized when cos θ =(r/R) 4.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 53.Continued.

Example, Page 265 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 59.Find the maximum length of a pole that can be carried around a corner joining corridors of widths 8ft and 4 ft. See Figure 30.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.7: L'Hopital's Rule

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In the case of the limit of a rational function at a point where the rational function has an indeterminate value, 0/0 or ∞/∞, we may use L’Hôpital’s Rule to find the derivative.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 277 Show that L’Hôpital’s Rule is applicable and use it to evaluate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company