Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: (75) Next: Homework Assignment 19 Previous: Term Exam 3

# Solution of Term Exam 3

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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 :

Solution. (Graded by Cristian Ivanescu) Let be an arbitrary partition of . Then for any the infimum is 0 and so . Thus . At the same time, for any the supremum is , and hence and so . Now let be given and let be the partition . Then while and so . Thus . But this is true for any and we already know that . So it must be that . Thus and hence is integrable on and its integral is .

Problem 2. Prove that the function

is a constant function.

Solution. (Graded by Julian C.-N. Hung) Differentiate using the first fundamental theorem of calculus. The first summand yields . The second summand is the first summand pre-composed with the function . So by the chain rule, the derivative of the second summand is . is the sum of these two terms, . Hence is a constant.

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 .

Solution. (Graded by Julian C.-N. Hung) Let be a constant and consider the functions and . Then and so both and satisfy . We also have and . So by what we have proven in class or .

Problem 4. The function is defined by .

1. Compute for all .
2. Explain why has a differentiable inverse for .
3. Let be the inverse function of (with the domain of considered to be ). Compute and simplify your result as much as you can. Your end result may still contain in it, but not , or .

1. .
2. For we have that and hence and . So is increasing on . It is also differentiable on that interval, so by a theorem proven in class, it has a differentiable inverse.
3. .

The results. 82 students took the exam; the average grade was 69.3, the median was 78 and the standard deviation was 26.76.

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Dror Bar-Natan 2004-10-18