Consider the function f of one variable with equation y = f(x) = ln(a - e^) where a is an unknown real number. a) Find the value of a for which the instantaneous rate of change at x = ln 2 is equal to -0.4. Using the value you found in a), find the domain of the function f and show that the function is decreasing everywhere on its domain. b) c) Find all horizontal and/or vertical asymptotes of the function f using the value you found in a).
Consider the function f of one variable with equation y = f(x) = ln(a - e^) where a is an unknown real number. a) Find the value of a for which the instantaneous rate of change at x = ln 2 is equal to -0.4. Using the value you found in a), find the domain of the function f and show that the function is decreasing everywhere on its domain. b) c) Find all horizontal and/or vertical asymptotes of the function f using the value you found in a).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the function f of one variable with equation y = f(x) = ln(a - ex) where a is an unknown real
number.
a) Find the value of a for which the instantaneous rate of change at x = ln 2 is equal to -0.4.
Using the value you found in a), find the domain of the function f and show that the function is
decreasing everywhere on its domain.
b)
c) Find all horizontal and/or vertical asymptotes of the function f using the value you found in a).
d) Based on the results from b) and c), sketch the graph of the function f.
e)
A function that is its own inverse is called an involution. Based on the graph you made in d), explain
why this function f is an involution.
Next, assume x = p (price) and y = q (demand) for a function of the same type as f. Hence, we define the
demand function g by
q = g(p) = ln(b - ep)
where b is again an unknown real number.
f)
Find the value of b if you know that the demand has unit elasticity when price p = ln 6. You are
allowed the use of your graphical calculator to find the zeros of a function if needed.
g)
Using the value you found in f), find a quadratic approximation of the demand function for values of
the price close to In 10.
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