46 EXTRA CREDIT PROBLEMS FOR MATH 75 / WARREN D. SMITH / FALL 2002. =================================================================== YOU CAN CHOOSE ANY PROBLEM OR PROBLEMS FROM THIS LIST! [SOME PROBLEMS ARE MULTIPART a,b,c, etc] DO THEM (PLEASE LABEL SAYING WHICH PROBLEM # IT WAS) & HAND THEM IN & I'll GIVE YOU EXTRA CREDIT! (Show all work. Warning, I may not award partial credit if answer is wrong. Also warning, if answeris right but I can't tell how you did it, that also may not get you credit. But, the bottom line is, you are not going to pay if you try these, you can only win.) Typewriter Notation: sqrt(x) = SquareRoot(x) ------------------------------------------------------ PROBLEM 1. Differentiate (d/dx) these (regard all letters besides x as constants): (a) sqrt(sqrt(x)+x) (b) sqrt((1+x)/(1-x)) r s (c) (1-x) / (1+x) 2 2 (d) x/sqrt(a - x ) PROBLEM 2. Differentiate (d/dx) these: 2 x (a) (x +1) 2 (b) the integral from 0 to x of exp(-s ) ds 2 2 (c) the integral from 0 to x of exp(-s ) ds [Hint: chain rule & Fund Theorem of Calc] cos(x) (d) x x+cos(x) (e) 3 PROBLEM 3. What is the area of the region of the XY plane with Y>0, X>1, and 2 X Y < 1 ? Note: this region is infinitely long, but, does that force its area to be infinite? PROBLEM 4. Can you prove that 2 3 4 5 6 7 8 1 + x + 2 x + 3 x + 5 x + 8 x + 13 x + 21 x + 34 x + ... 2 = 1 / ( 1-x-x ) ? What is the rule that governs the sequence 1,1,2,3,5,8,13,21,34,...? PROBLEM 5. Let F(x) = G(H(x)) i.e. the composition of the functions G and H. (a) What is F''(x)? (b) What is F'''(x)? PROBLEM 6. A 20-foot-long ladder leans against a wall. You move the base of the ladder away from the wall at speed 1 foot per second. At what speed is the top of the ladder sliding down the wall at the moment when it is 12 feet above the ground? PROBLEM 7. A ladder L feet long leans against a high wall, and it goes over, without touching, a fence located 5 feet in front of the wall and which is 10 feet high. What is the minimum possible value of L? PROBLEM 8. Show that if X is an integer with X>24, then the square root of X and the square root of X+1 differ by less than 0.1. PROBLEM 9. Suppose the nth derivative (n primes in all) of F(x) is 0. Describe the full set of functions F(x) that work. PROBLEM 10. A triangle has legs 3 and 4 (in meters), and the angle between them increases at a rate of 1 degree (i.e. pi/180 radians) per second. At the moment when the triangle is a right triangle (a) What is the hypotenuse? (b) At what speed (in meters/second) is that hypotenuse increasing in length? PROBLEM 11. Suppose you have a 1x1 square piece of paper. You cut away four KxK subsquares (for some K with 00. (Hint: differentiate both sides and use an idea like the "racetrack principle".) PROBLEM 17. Find a simple relationship between arctan(x) and arctan(1/x) [hint, try a few values with calculator first - see a pattern?], then prove it. PROBLEM 18. Consider the following "all purpose volume formula". Volume = height * [A0 + 4 Ahalf + Afull] / 6, where A0 = area of cross section at zero height, Ahalf = area of cross section at half height, Afull = area of cross section at full height. For example, this claims that the volume of a cone of height H and radius r is H * [pi r^2 + 4 pi (r/2)^2 + pi 0^2] / 6. Does this formula work for (a) a sphere? (b) a cone (height measured along axis)? (c) a cylinder (height measured along axis)? (d) a box? (e) a pyramid? Can you think of another shape, for which this formula does NOT work? PROBLEM 19. If i is the square root of minus 1 (imaginary number, discussed in an appendix C of the book) then what is i arctan(i)? Hint: consider the Taylor series of arctan(x). Also, consider the Taylor series of ln(1+z). PROBLEM 20. Find the equation of a circle whose perimeter passes thru the points (1,1) (3,2) and (0,6). PROBLEM 21. Suppose, on a day in which the outside temperature is 0 Celsius, the rate (in degrees Celsius per second) at which the center of a hot ball of iron cools, is T'(t) = -T(t) / r where r is the constant radius of the ball in cm, and T is measured in Celsius. (And t is the time, measured in minutes). Note this law would have the property that the hotter the iron is, the more quickly it cools off; the thicker it is, the more slowly it cools. Can you determine the amount of time it takes for a ball of radius 10cm to cool from central temperature 100 C down to 1 C? (Does your answer sound roughly correct?) Now determine the amount of time it takes for the entire earth (modeling it as an iron ball) to cool from central temp 5000 C down to 1 C? (It probably will help if you find a formula for T(t) in terms of t, the initial temp T(0), and the radius r. To derive such a formula, start by asking: what function of t, if you take its derivative, yields the same function back, only scaled by 1/r?) This kind of calculation was made in the late 1800s and leads to a wildly incorrect estimate of the age of the Earth. The reason for the erroneousness, it was later realized, is that radioactive atoms inside the Earth generate heat, which this cooling-only calculation (made before radioactivity was discovered) had ignored. PROBLEM 22. Suppose a bird flies along a line at speed 5 meters/sec. At its closest approach to you, the bird is directly in front of you and 4 meters away. You turn your head so that you are always looking directly at the bird. At what rate (in radians per second) are you turning your head 1 second later? PROBLEM 23. (a) Solve the equation x=cos(x) for x. I want an answer accurate to 7 or more decimal places. (b) Solve x+ln(x) = 0 for x. I want an answer accurate to 7 or more decimal places. PROBLEM 24. Describe the full set of x-intervals in which the graph of y=sin(x) is concave-up? PROBLEM 25. Describe the full set of x-intervals in which the graph of y=tan(x) is concave-up? PROBLEM 26. Describe the full set of x-intervals in which the graph of y=exp(sin(x)) is concave-up? PROBLEM 27. You are standing in a wide open flat field on a sunny day. You have a meter stick but nothing else. There is a tall redwood tree in the center of the field. It is far too high to climb or to throw a stone that high. And no, do not chop it down. How can you, easily, determine the height of the tree accurate to about 1%? PROBLEM 28. Suppose you have 3 beams, each 10 meters long. You attach the beams (possibly at some bent angles) at their ends 3 in a row, and attach the ends of two outer beams to the ground, creating a quadrilateral "arch" (the 4th side of which, is the ground). What is the maximum area this arch can have? PROBLEM 29. 2/3 2/3 Consider the curve defined by x + y = 13. What is the equation of the line that is tangent to this curve at the point x=8? (And what is y at this x - assume we know y>0)? PROBLEM 30. You and your friend are both standing on a planet. The sun is directly above you. You radio your friend and tell him this. He says he is standing next to a vertical pole 10 meters high, which is casting a shadow 5 meters long. Your friend is 100 km North of you. WHAT IS the diameter of the planet? (Note: The ancient Greeks knew the Earth was round, and they found out its size by a method similar to this. Unfortunately this was forgotten during the "dark ages" in Europe, causing Columbus to have a difficult job convincing people the Earth was round, and also causing Columbus to have a wildly incorrect estimate of the Earth's size! Columbus would have died if America had not been there... he got lucky...) PROBLEM 31. Find the following limits (all as x --> 0): 3 2 (a) lim [ x + 5 x + tan(3x) + sin(x )/tan(5x) ] sin(9x) / [1 - cos(x)] 3 (b) lim arctan(3sin(x)) / tan(x + 3x + sin(x)) (c) [For this define the sign function by sign(x)=1 if x>0, sign(x)=-1 if x<0, and sign(x)=0 if x=0.] lim |sign(x)| PROBLEM 32. 4 3 (a) Find y'(x) in terms of x and y if x + y = 18 x y (b) Same thing, but x + sin(y) = x/y PROBLEM 33. x y (a) Find y'(x) in terms of x and y if x + y = 1 (b) Same thing, but x cos(y) + sin(y) / x = xy PROBLEM 34. Suppose you want to design a cylindrical open-topped can to hold 1 liter of water. (1 liter = 10cm X 10cm X 10cm.) What is the least amount (i.e. area) of metal sheet you need (including both the cylindrical curved side wall, and also the circular flat base)? What dimensions (height and radius) does the can have? PROBLEM 35. Find the dimensions (height and base-radius) of the largest-volume cone that will fit inside a sphere of radius 1. What is its volume? PROBLEM 36. Water runs into a spherical tank of radius 1000 cm at a rate of 1 liter per second. (1 liter = 1000 cubic cm.) When the water inside the tank has risen to height 700 cm, at what speed (in cm/second) is the water surface rising? PROBLEM 37. Find the derivative (d/dx) of the following: (a) integral from 0 to sqrt(x) of sin(t) d t ln(x) (b) x PROBLEM 38. (a) What is the quadratic approximation to cos(x - pi/4) based at x=0? (b) What is the quadratic approximation to ln(1 + exp(x)) based at x=0? PROBLEM 39. Find these limits as x --> 0: sin(x) x (a) ( 2 - 1 ) / ( e - 1 ) (b) x ln x PROBLEM 40. Rank the following 12 things in increasing order of size in the limit as x --> 0+: 2 2 3 x 1/x -1/x x (x ) -1/2 1/2 x x |ln(x)| x x|ln(x)| 2 3 3 6 5 x x PROBLEM 41. Same problem, but as x --> +infinity instead. PROBLEM 42. find lim as x--> +infinity of: (a) cos(1/x) / (1 + 1/x) (b) (sin x) / (2 x + 3) (c) (2x + sin(x))/x 199 x (d) x / 3 PROBLEM 43. (a) Suppose your salary increases every year by 3 percent. Initially it is $100. After 30 years, what is it? (b) What is the current population density (people per square kilometer) on the surface (including oceans) of the Earth? (c) Suppose the population of the Earth doubles every 40 years. (Which has been about true over the last 200 years.) How many years until the population density reaches 1 person per square meter? PROBLEM 44. A function F(x) has ODD symmetry if F(x)+F(-x)=0 and EVEN symmetry if F(x)=F(-x). (Also: It could have neither.) What kind of symmetry does the derivative F'(x) have, if F is odd? And if F is even? PROBLEM 45. (You almost certainly will need to go to a library and look in a table of integrals for this one; the idea is to make you have that experience. In math76 you will learn how to do these without needing a table of integrals.) What are the following integrals: 2 (a) integral from 0 to x of 1 / sqrt(5 x + 7 x + 99) dx (b) integral from 0 to x of tan(x) + sin(x) dx (c) integral from 0 to x of x^3 sin(x) dx PROBLEM 46. Suppose $F(y)$ is the inverse function of $G(x) = \sin ( \exp(x) )$ (defined when $x<0$). Write a formula, solely in terms of y, for the derivative $F'(y)$ of $F(y)$? ===================================================================