136.169, Assignment No. 4

January 17, 2003

The assignment is due Friday, January 24, 2003 in class. Late assignments receive a mark zero.

1. Evaluate the limits :

a) . [3]

b) . [3]

c) . [3]

2.Find the n-th derivative of f(x) = ln (x+1). Prove the formula by using induction.[5]

3. Let f(x) = ln (x+1).

a)Write the Taylor polynomial Pn(x) and the Lagrange remainder Rn(x) for

f(x) around =0 . [3]

b) Show that for x=1. [3]

c) Find n such that the approximation of ln2 by the n-th Taylor polynomial Pn

for ln (x+1) around 0 is with an error smaller than 0.001. [4]

4.a) Prove that a general cubic polynomial ,

has exactly one inflection point . [5]

b) Prove that if a cubic polynomial has three x-intercepts, then the inflection point

occurs at the average value of the three intercepts. [4]

5. Draw the graph of the function by showing all the

details of your work. [9]

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Total [42/40]