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Unformatted text preview: Homework 9 - Math 139
Due Nov. 3rd Instructor
Web page Mauro Maggioni
293 Physics Bldg.
www.math.duke.edu/˜ mauro/teaching.html Reading: from Reed’s textbook: Section 4.3,4.5,4.6
§4.2: #2, 3, 4, 5, 6, 12
§4.3: #7, 11(a,b,d,e)-without using Taylor’s Theorem
1. Examine the diﬀerence quotient used in the deﬁnition of the derivative of cos x and
write down, but do not evaluate, the limits you need to know in order to compute cos′ x.
In this context, what is wrong with your answer to # 4, §4.2?
2. Use Taylor’s theorem to show that if f and g are (n + 1)-times continuously differentiable functions on an open interval containing x0 , f (k) (x0 ) = g (k) (x0 ) = 0 for
k = 0, 1, . . . , n, and g (n+1) (x0 ) = 0, then
f (n+1) (x0 )
x→x0 g (x)
3. Use 2. to compute
x2 − sin2 x
x→0 x2 sin2 x
4. In #11(a) above you showed that ln x → ∞ as x → ∞. Use a similar argument to
as n → ∞. (Pick an appropriate partition P of [ n , 1] and show that the corresponding
lower sum tends to ∞ as n → ∞.) Conclude that ln x → −∞ as x → 0. ...
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This note was uploaded on 01/16/2012 for the course MATH 139 taught by Professor Pardon,w during the Fall '08 term at Duke.
- Fall '08