MATA02H3 Lecture Notes - Lecture 2: Euclidean Algorithm, Multiplication Table

Show all work No late acceptance!! Solve all problems using the methods learned in chapters 2.9,3.4, 3.5 &3.8 I will not accept answers that do not show calculus solutions la. Determine the local linear approximation of (x)sin(3x) near x-0 lb. Approximateusing the method of local linear approximation (2.05) 2. Determine the extrema for the following functions over the indicated intervals: 3. a) Determine whether Rolle's Theorem applies to the following function on the given interval. If so, find the point(s) that are guaranteed to exist by the conclusion of the theorem. f(x)-1-r2/3 1,1] b) Determine if MVT applies to the following function on the given interval. If so, find the find the poin(s) that are guaranteed to exist by the conclusion of the theorem. f(x) =2x-- [3,6] The surface area of a cube is to be calculated from the measured value of its side, what is the maximum possible percentage error in the measurement if the percentage error in the calculation of the surface area is to be kept within ± 1.5% 4- The hypotenuse of a right triangle is measured to be 12 cm with a possible error of ±1/2 cm. a) What is the error in the calculation of the angle θ, if the side adjacent to θ is known to 5. be exactly 6 cm? b) What is the maximum possible percentage error in the measurement of the adjacent side and the calculation of If fertilizer made from N pounds of Nitrogen and P pounds of Phosphate is used on an acre of farmland then the yield of com (in bushels per acre) is given by the following: Y =75 + 6N +7 P-N2-2p2 + NP If nitrogen costs $0.25/lb and phosphate costs $0.20/lb, which combination of N and P produces the highest yield of corn if the farmer intends to spend $30/acre? 6. A closed rectangular box with a volume of 18000 in' is to be made so its base is a rectangle with length 3 times its width. Find the dimensions that will minimize the surface area. 7.

Math 151 - Sections 34, 35, 36 - Workshop 11 This workshop deals with an addition formula in section 5.1. This formula is related to an ld story about the great mathematician Gauss. Here is one version of the story Wh en Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math lecture period. To keep the students busy while he graded, he gave them the following task at the start of the period: Use your slate (the small personal blackboard that every student had in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate and yelled "Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the period Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer Then Gauss told everybody that the sum 1+2+3+ + 100 is ( 1 + 100) + (2+99)+ (2+98)+--+ (50+51) = 101 + 101 + 101 +. . . +10 101. 50-5050 Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prov the formula 1 +2+3+.+N) be an even number Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a+1.b) + (a+2 b) + (a+3.b)+. + (a + N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+(a+2.b)+ (a +3.b)+(a+ N b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this ve clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3++N, he could simply add N+1 to that sum, and that would give him the sum 1+2+3 Now assume that, by trial and error, Gauss discovered that the formula 1+2+3+ +N an even numbe +) Include the reason why it is important that +··· + (N +1) very quickly is true for several values of N. How could he prove that the formula is really true for ALL N? The insight described in the previous paragraph might have led Gauss to think the following: All he had to do was prove N(N+1) ± (N + 1) other words, he only needed to show: Formula for 1 + 2+3+.+ N plus N +1 equals formula for 1+2+3+(N+1). He would have discovered the principle of mathematical induction Problem 5. Prove of the formula for 1+2 +3+... + N (N +1)(+1X++1) Show how this leads to a proof N(N+1 N+)