Name(s): Score:
Math 148 Lab Assignment 3: § 7.4-7.6
Directions: You may work in groups of 2–3 to complete this assignment. Answer each question completely. Show all work to receive full credit, and circle your final answer.
1. Determine whether each integral is convergent or divergent.
(a)
Z
∞
0 x
( x 2 + 2) 2 dx
(b)
Z
6
−∞ xe x/ 3 dx
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2. Consider the function f ( x ) =
√
3 x .
(a) Find the second-degree Taylor polynomial for f at a = 8.
(b) Taylor’s Inequality: If | f ( n +1) ( x ) | ≤ M , then
| R n
( x ) | ≤
M
( n + 1)!
| x − a | n +1
.
Use Taylor’s Inequality to determine the maximum error for the approximation f ( x ) ≈ T
2
( x ) for 7 ≤ x ≤ 9.
2

3.
Logistic Growth Let N ( t ) denote the population size of a species at time t ≥ 0 and suppose that the population exhibits logistic growth. That is, dN dt
= rN 1 −
N
K where r and K are positive constants.
= f ( N ) ,
(a) Find the first-degree Taylor polynomial for f ( N ) at a = 0.
(b) Use your result in part (a) to explain why, for small population sizes ( N ≈ 0), the population exhibits near-exponential growth. That is, dN dt
≈ rN.
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