Investigation Consider the function f ( x ) = 2 x n x 4 + 1 for nonnegative integer values of n . (a) Discuss the relationship between the value of n and the symmetry of the graph. (b) For which values of n will the x-axis be the horizontal asymptote? (c) For which value of n will y = 2 be the horizontal asymptote? (d) What is the asymptote of the graph when y = 2 (e) Use a graphing utility to graph f for the indicated values of n in the table. Use the graph to determine the number of extrema M and the number of inflection points N of the graph. n 0 1 2 3 4 5 M N
Investigation Consider the function f ( x ) = 2 x n x 4 + 1 for nonnegative integer values of n . (a) Discuss the relationship between the value of n and the symmetry of the graph. (b) For which values of n will the x-axis be the horizontal asymptote? (c) For which value of n will y = 2 be the horizontal asymptote? (d) What is the asymptote of the graph when y = 2 (e) Use a graphing utility to graph f for the indicated values of n in the table. Use the graph to determine the number of extrema M and the number of inflection points N of the graph. n 0 1 2 3 4 5 M N
Solution Summary: The author explains how to determine the relation between the symmetry of the function f(x) and the value of n.
(a) Discuss the relationship between the value of n and the symmetry of the graph.
(b) For which values of n will the x-axis be the horizontal asymptote?
(c) For which value of n will
y
=
2
be the horizontal asymptote?
(d) What is the asymptote of the graph when
y
=
2
(e) Use a graphing utility to graph f for the indicated values of n in the table. Use the graph to determine the number of extrema M and the number of inflection points N of the graph.
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and
use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three
investment?
STEP 1: The formula for compound interest is
A =
nt
= P(1 + − − ) n²,
where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to
A = Pert
Find r and n for each model, and use these values to write A in terms of t for each case.
Annual Model
r=0.10
A = Y(t) = 1150 (1.10)*
n = 1
Quarterly Model
r = 0.10
n = 4
A = Q(t) = 1150(1.025) 4t
Continuous Model
r=0.10
A = C(t) =…
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