Math 1180 – Turn-in Homework Project - 50 pts. Due date: Thursday, May 4, 2017 Part 1 – Math proof problems 1. (5 pts.) Recall the definitions of two important number sets: p   The rational numbers =  p and q are integers, q  0 q   The irrational numbers is the set of numbers that are not rational (e.g.,  , 2 , e, etc.) Consider the following function:  x, if x is an irrational number f ( x)   0, if x is a rational number Prove that f is discontinuous at every irrational number. Hint: You may want to use the fact that every nonempty open interval of real numbers contains both rational and irrational numbers.

2. (6 pts.) Consider the following function: 1  2  x sin if x  0 f ( x)   x  0 if x  0 a) Prove that f is differentiable at x = 0. b) Prove that f  is not continuous at x = 0.

3. (5 pts.) Suppose that a function f (x) satisfies the following conditions for all real values of x and y: a) f ( x  y)  f ( x)  f ( y) b) f ( x)  1  xg ( x) where g (x) is a function with lim g ( x)  1 x 0

Prove that f (x) exists at every value of x and that f ( x)  f ( x)

4. (5 pts.) Let f (x) be any continuous function defined on the interval [0, b]. Prove that b f ( x) b  0 f ( x)  f (b  x) dx  2 b b f ( x) f (u ) Hint: Let A   dx and A1   du . Note that A1  A . 0 f ( x)  f (b  x) 0 f (u )  f (b  u ) In the A1 integral, make the change of variables u  b  x and let A2 equal the resulting integral. Note that A2  A1 . Then, combine two of these integrals in some way to determine the value of A.

5. (5 pts.) A famous integration approximation technique states that for any continuous function f (x)

 For example,



1

1 1

 1  f ( x)dx  f     3 

e x dx  e  e 1 2.350402387 and e 1 /

1

3

 1  f    3

 e1 /

3

 2.342696088

Prove that this approximation is exact for all polynomial functions, f (x) , of degree 3 or less.

Part 2 – Application problems 6. (6 pts.) Everyone knows that the area of a circle of radius r is  r 2 . But, if you happened to live before 300 B.C., you would not have known this formula for the area of a circle. It was Archimedes who derived the area of the circle, around 250 B.C. Here is a method similar to the one he used. Let n be a positive integer and cut the circle into n equal sectors. In each sector there is an isosceles triangle formed by the center of the circle and the points where the straight edges of the sector intersect the circle. He found the area of each triangle and then approximated the area of the circle by the sum of the areas of the triangles. Finally, he let the number of triangles (n) increase without bound to find the area of the circle. Your goal is to repeat this procedure and prove that the formula for the area of a circle is correct. It may help you to figure out the central angle (the angle at the center of the circle) of each sector. Unlike Archimedes, you may make use of trigonometry to assist in calculating the sum of the areas of triangles. What happens to this angle as the number of triangles increases without bound? What happens to the sum of the areas of the triangles? Since your answer agrees with the circle area formula that we already know, you probably think that you have found the area of the circle. But, you must be sure that the process above does not “miss any of the area”. To check this, you should also estimate how far off your approximation of the area of the circle is and show that this error approaches zero as the number of triangles increases without bound. Divide the circle into n sectors and form the inscribed triangles as before. At the base of each isosceles triangle, form the smallest rectangle that contains the area of the sector that is outside of the triangle. What is the area of this rectangle? Use this to bound the error in approximating the area of the sector by the area of the triangle. Find a bound for the total error in approximating the area of the circle by the sum of the area of the triangles (i.e., the sum of the areas of the rectangles). Show that this error approaches zero as the number of triangles increases without bound – this will then prove that your formula for the area of the circle is correct.

7. (6 pts.) The star nearest our sun is Alpha Centauri, which is about 4 light years from earth. Alpha Centauri is so far away that when its light reaches earth, it is traveling in essentially parallel rays. To observe distant stars, astronomers use mirrors shaped like paraboloids, which are parabolas rotated about their axes of symmetry. The reason they use a paraboloidal mirror is that it focuses all the light to a single point, the “focus.” (This point is the image of the star in the paraboloidal mirror.) In solving this problem, you will demonstrate the focusing property of parabolas. a) Suppose our mirror is shaped like the parabola f ( x)  kx2 , where k is any positive constant. Find the coordinates of its focus in terms of k (see Sec.10.5 of our text for a review of the conic sections definition of a parabola and its focus). b) Find the equation of the line tangent to the parabola at a point (a, f (a)) on the parabola. Then, find the y-intercept of the tangent line. c) Suppose that an incoming light ray strikes a curve at the point (a, f (a)) on the parabola. If the light ray makes an angle  with respect to the tangent line, then it is reflected at an equal angle to the tangent line. This result from physics is known by the phrase “the angle of incidence equals the angle of reflection.” Using this fact, argue that the incoming light rays parallel to the axis of the parabola are all reflected to the focus, independent of the point of incidence. Thus, a parabolic mirror focuses incoming light rays parallel to the axis to a point.

Hint: Consider the triangle with vertices at the point of incidence, (a, f (a)) , the y-intercept of the tangent line found in part (b), and the point (0, b) on the y-axis where the line of reflection of the light ray intersects the y-axis. Use the physics fact from part (c) to prove that this triangle is isosceles (this requires some geometry). Then, using the result that the triangle is isosceles, prove that the (0, b) is the focus of the parabola.

8. (6 pts.) You have a sheet of paper that is 6 inches wide and 25 inches long, placed on a table so that a short side is facing you. The top corners and the lower left corner are taped down. Fold the lower right corner over to touch the left side. Your task is to fold the paper in such a way that the length of the crease is minimized. What is the length of the crease? Give both the exact answer and the answer rounded to the nearest 0.001. 9. (6 pts.) George and Martha have decided to add a greenhouse to their house. Their plan is to knock out the lower portion of the entire length of the back wall of their house and turn it into a greenhouse by replacing the removed lower portion of the wall by a huge rectangular piece of glass placed at an angle to the wall between the wall and the ground. They have also decided that they are going to spend a certain fixed amount of money on this project. Consequently, the cost of the glass to be used is fixed. The triangular ends of the greenhouse will be made of various materials that they already have lying around the house (and so the ends will not add to the cost of the project). The floor space in the greenhouse is only considered usable if they can both stand up in it, so part of it will be unusable, but they don’t know how much. Of course this depends on how they configure the greenhouse. They want to position the glass top of the greenhouse to get the most usable floor space in it, but they are at a real loss as to how to do this and how much usable floor space they will get. They ask you for help in solving this problem. George and Martha forgot to give you any of the relevant specific numbers (e.g., dimensions, the fixed project cost, etc.) But, after some thought, you realize that you don’t need any of the relevant dimensions to solve the problem. Your job is to determine how the glass should be positioned so that the usable floor space in the greenhouse will be maximized, and to determine the maximum usable floor space. Note: Your answers will not be numbers, but will consist of two formulas – one for the angle the glass top makes with the ground in terms of constants, and a second for the maximum usable floor area in terms of constants and the angle. When the relevant numbers become available, your formulas could be used to obtain actual number answers. Your answer should also clearly identify all constants and variables that appear in your formulas. Hints: See the figures on the next page. Identify the usable floor space and all relevant quantities and label them with letters. Then, identify which of the quantities can be considered constants for this problem and which are variables. For example, the length of the back of the house is fixed and is a constant. On the other hand, the angle that the glass top of the greenhouse makes with the ground is a variable.

George and Martha construction project figures:

current back view of the house

back view with greenhouse portion of back wall removed glass top

What you need to turn in: Write the details of your solutions with supporting explanations for all the problems. Clearly identify the problem number for each solution. Write your name on every solution page. Put your solutions in order based on problem numbers. Put a copy of this project assignment in front of your solutions. Staple all your pages together. Note: Solutions with insufficient support or with problems in a jumbled order or turned in unstapled will lose points. Working in Pairs: If you want to work with one other student in our class on this assignment, that is acceptable provided that both members of the pair make a contribution to the solution. If you decide to work in a pair, turn in only one copy of the solution – clearly write the name of each pair member on your solution.