|Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:||(74)||
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Solve the following 4 problems. Each is worth 25 points although they may have unequal difficulty. Write your answers in the space below the problems and on the front sides of the extra pages; use the back of the pages for scratch paper. Only work appearing on the front side of pages will be graded. Write your name and student number on each page. If you need more paper please ask the tutors. You have an hour and 50 minutes.
Allowed Material: Any calculating device that is not capable of displaying text.
Problem 1. In a very condensed form, the definition of integration is as follows: For bounded on and a partition of set , , and . Then set and . Finally, if we say that `` is integrable on '' and set .
From this definition alone, without using anything proven in class about integration, prove that the function given below is integrable on and compute its integral :
Problem 2. Prove that the function
Problem 3. In class we have proven that a twice-differentiable function satisfying the equation is determined by and . Use this fact and the known formulas for the derivatives of and to derive a formula for in terms of , , and .
Problem 4. The function is defined by .