217-zhu-07sp02 - MATH 217 MIDTERM2(SPRING 2007 N ame...

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Unformatted text preview: MATH 217: MIDTERM2 (SPRING 2007.) N ame:_______-__-___---___________-______- Section: -____- _____ TA:-___-_____________ Score: Problem 1.______r________-_____-l____ Problem 2 ____________________________ Problem 3 ____________________________ Problem 4.n---___-_________._____T__ Problem 5 ____________________________ Problem 6 Total:___-_r _____________________ Instruction: Show all work. No work = no credit, even if you have a correct answer. References and calculator are not allowed. ‘ Problem 1 (10 points): Find the volume of the solid generated by. revolving the region between the x-axis and the curve 3/ =_%, 1 S a: S 4, about the . x-axis. " V 20:4- Problem 2 (10 points): The function f(a:) = 3953;; is defined on {0,+oo). Show that f is one to one. Then find the expression of its inverse function. (You don’t have to specify the domain and range of its inverse function). Problem 3 (10 points): Use logarithm differentiation to compute f’ (x), where f(:z:) = (2x + 1F” , cc > —%. Then evaluate f’(0). , Problem 4 (20 points): Compute the following: (a) % Ink-953%) . Problem 5 (10 points): (a) Show that any two logarithm functions logaa: and logbx (here a, b > 1) have the same growth rate as a: ——> +00. (b) Show that :02 + sinx is of smaller order than 2x as cc —> +00, in other words, 3:2 + sinac = 0(2‘”). Problem 6 (10 points) Evaluate the following: (a) tan(sin‘1(%)) ' (b) cos—1(cos(——§—)) ...
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