Name Exam 1
Math 105, section D October 11, 2002
1. (12) 3. (22) 5. (22)
2. (26) 4. (18)
Total
1. (12 points) If f(x) = ln (ex), calculate f0(x). Simplify your answer if possible.
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The math 105 exam 1 for section d, held on october 11, 2002. The exam covers various topics in calculus, including differentiation and integration. Questions include finding derivatives of functions, determining continuity and calculating limits.
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Name Exam 1 Math 105, section D October 11, 2002
- (12) 3. (22) 5. (22)
- (26) 4. (18)
Total
- (12 points) If f (x) = ln (ex), calculate f ′(x). Simplify your answer if possible.
- (26 points) Suppose g(x) =
x^2 − 5 x + 6 if x ≤ 1 4 − x − x^2 if 1 < x ≤ 2 x^2 − 3 x if x > 2.
(a) Is g(x) continuous at x = 1? At x = 2? Explain. (b) Calculate g′(x). Does g′(1) exist? Does g′(2) exist? Explain. (c) Calculate g′′(x). Does g′′(1) exist? Does g′′(2) exist? Explain.
- (18 points) Use the fact that d dx
|x| = |x| x
, and whatever other derivative rules you may need, to
calculate the following derivatives. Simplify your answers if possible.
(a) a(x) = |x| x−^1 (b) b(x) = ln |x|
- (22 points) Calculate the derivatives of the following functions. Do not try to simplify your answers!
(i) c(x) =
x^11 + 7x^4 − 6 x−^2
x^8 + 8 + x−^8
(ii) d(x) = ec(x), where c(x) is the function in (i). (Use your answer to (i) in this answer if you like.)