Advanced Calculus Questions

Transcription

1 Advanced Calculus Questions What is here? This is a(n evolving) collection of challenging calculus problems. Be warned - some of these questions will go beyond the scope of this course. Particularly difficult questions will be indicated with. How should I use it? We should only be attempting these problems when we feel comfortable with the definitions, ideas and problem solving techniques at the level of the homework. Learning to walk before learning to run, and all that. If, however, you feel comfortable with the homework problems and are still thirsty for some more calculus - then dive in! Don t worry if you can t answer everything right away - that s not the point. Contents 1 On the limit of x x at ( ) An actual definition of the limit 3 3 Fixed point on a elastic band 4 4 Antipodal temperatures 5 5 Area of a circle 6 6 The integral of an exponential via Riemann sums 7

2 1 ON THE LIMIT OF X X AT On the limit of x x at 0 On the one hand we convinced ourselves that x 0 = 1 for all real numbers x such that x = 0. On the other hand, we saw that 0 x = 0 for any x > 0. So how should we think about the 0 0? Let s investigate the following limit, lim x 0 + xx 1. Let s first plug in a sequence of values 1, 1/2, 1/3, 1/4 etc. on a calculator and guess what is happening. 2. Let f(x) = x x and consider the sequence a n = 1/n (which converges to 0 from the right). Try convince yourselves that lim n f(a n ) = Have we found the limit then? If not, why not, and what can we conclude from this statement? 4. (a) (Without L Hôpital s Rule) For y (1, ) we have that 0 < ln y < y. By setting y = x 1/2 show that, lim x ln x = 0. x 0 + [Hint: what happens to y = x 1/2 as x?] (b) (With L Hôpital s Rule) Use L Hôpital s Rule to deduce that 5. Deduce that lim x 0 + x x = 1. lim x ln x = 0 x 0 + (*) Consider the same question only now with f(x) = x xx. What do you think the limit as x 0 + is now? xx x (**) Consider the same question only now with f(x) = x a tower of powers of x! What do you think the limit as x 0 + is now?

3 2 ( ) AN ACTUAL DEFINITION OF THE LIMIT 3 2 ( ) An actual definition of the limit Here is the precise definition of the limit. Definition 2.1. We write, lim f(x) = L, x a if the following condition holds. ɛ > 0, δ > 0 such that, for all x, x a < δ f(x) L < ɛ. 1. Give the correct formulations for the left and right hand limit in the form of this precise definition. 2. Give the correct formulation of the definition of continuity in the form of this precise definition. 3. Show that f(x) = x + 4 is continuous at a. 4. Suppose that f(x) and g(x) are continuous at a, show that (f + g)(x) is continuous at a. 5. Prove that sin(1/x) does not have a limit at 0.

4 3 FIXED POINT ON A ELASTIC BAND 4 3 Fixed point on a elastic band Consider an ordinary elastic band. In this question we ll try to convince ourselves that when we stretch it there must be at least one fixed point! We will model this situation as a function f : R R and apply the intermediate value theorem. To make life easier, think of the elastic band not as a circle, but rather as an interval. Suppose that to begin with our elastic band has length 1, we grip either end of the band and stretch it - pulling the left end further to the left, and the right end further to the right, until it has length α > Describe a function f : [0, 1] R that models this situation. You don t need to give an explicit formula for f. 2. Investigate the behaviour of f at it s endpoints. 3. Construct a new function d f (x) = f(x) x. Is it continuous? How does it behave at it s endpoints. 4. The functions f and d f have the same domain. Let c be a fixed point of f. What can we say about d f (c)? Is the converse true? 5. Apply the intermediate value theorem to d f to conclude that f must have a fixed point.

5 4 ANTIPODAL TEMPERATURES 5 4 Antipodal temperatures We say two points on a sphere are antipodal if they are diametrically opposite to one another. For example, the north and the south pole are a pair of antipodal points. In this problem we are going to argue that at any given moment in time, there always exists a pair of antipodal points on the Earth with the exact same temperature! 1. Consider the equator of the Earth. In this question we model the temperature along the equator by a continuous function. Do you think this is a fair assumption? 2. Let R denote the radius of the equator. Further let m denote the point on the equator where it is exactly midnight, and n the point where it is exactly noon. Note that m and n are antipodal points. m n Does it seems like a reasonable assumption that the temperature at n is greater than the temperature at m? 3. The temperature along the equator can be modeled by a function T(x) from [0, 2πR] to R. Explain why. 4. Consider the function Explain why this is a continuous function. 5. What can you say about (a) A(n) (b) A(m) { T(x) T(x + πr) x [0, πr] A(x) = T(x) T(x πr) x (πr, 2πR] Use the intermediate value theorem to argue that the function A(x) has a zero in the domain [0, 2πR], and explain why this completes the problem.

6 5 AREA OF A CIRCLE 6 5 Area of a circle In class we have been estimating the area under a graph with rectangles. In this question we will employ similar ideas to estimate the area of a circle with regular polygons. The idea is to circumscribe a regular polygon inside the unit circle. First a triangle, then a square, then a pentagon, and so on. Each time the area of the given polygon is an approximation to the area of the circle. Note, we could also inscribe the unit circle inside the polygon, giving another estimate to the circle s area. To compute the area of our circumscribed n-gon (this is the name we give to the polygon with n sides, e.g., a triangle is a 3-gon, a square is a 4-gon, and so on) we subdivide it into n triangles (see figure below) and compute their areas. 1. Write a formula for the area of one of the n triangles using the following figure as a guide. [Hint: you know the length of the hypotenuse and the angle indicated] 2. Use the fact that there are n such triangles composing the circumscribed n-gon to give an expression for its area A n. [Hint: this should look similar to a Riemann sum] 3. Take the limit as n to obtain the familiar formula for the area of a circle. [Hint: you will need to use that fact that lim x 0 sin x x = 1] 4. ( ) This limit, lim n A n gives a lower bound to the area of the circle. Construct a parallel argument that gives an upper bound, and show that it agress with the formula you know. 5. ( ) Where did π come from? How did it enter our formula? What would change if we had worked in degrees as opposed to radians?

7 6 THE INTEGRAL OF AN EXPONENTIAL VIA RIEMANN SUMS 7 6 The integral of an exponential via Riemann sums 1. Expand to give a formula for the sum (1 + x + + x n 1 )(x 1) n 1 x i i=0 2. Set up the n-th lower Riemann sum for the exponential function f(x) = e x on the interval [0, a] to show L n = a n 1 e k/n n 3. Writing c = e 1/n and using your answer to part 1, show that i=0 L n = (e a a/n 1) e 1/n 1 4. Make the substitution h = 1 n. Explain why we are now interested in the limit lim h 0 h e h 1 5. This may seem like a strange thing to do, but change gears for a minute and write down the definition of derivative of e x at x = Put everything together to compute the integral a 0 e x dx Note: you do not need to compute any other Riemann sums so long as you mention an appropriate theorem!