THE UNIVERSITY OF WESTERN AUSTRALIA

Taylor Series, Logarithmic and Exponential Function

 

 

 

Business School Economics Semester 2, 2016
Money, Banking, and Financial Markets (ECON3350)
Tutorial 4. Taylor Series, Logarithmic and Exponential Function Solve all questions. Questions with an asterisk (*) will be marked.
Taylor Series Approximation
A k-times differentiable function can be approximated by a Taylor series: f ( x) f ( a ) f (a)
f (a)
f (a)
( x a) ( x a) 2 ( x a) k
1!
2!
k! Here, f (a) is the function value at x = a, f ʹ(a) is the rate of change at x = a, etc..
Question 1*
a) Determine the function value and the first and second derivative of f (x) = x4 at x = 3. Then,
estimate the function value at x = 3.1, using a first-order and second-order Taylor series
approximation. How big is the error? Illustrate with a graph.
b) Determine the function value and the first and second derivative of f (x) = ex at x = 0. Then,
estimate the function value at x = 0.1, using a first-order and second-order Taylor series
approximation. How big is the error? Illustrate with a graph.
Question 2
Show that the change in the logarithm of a variable is approximately equal to a percentage change.
Hint: Use a first-order Taylor approximation of the logarithmic function.
Question 3
Show that ln(1 R) R for small R.
Hint: Use a first-order Taylor approximation around R = 1.
(This approximation is often used in financial economics. For an interest rate of 3%, the logarithm
of the gross return (1+R) is approximately equal to R: ln(1 0.03) 0.03. )
Question 4*
Show that the time derivative of the logarithm of a variable equals its growth rate measured in
percent. Use the following notation for a time derivative: dX / dt X . 1 Question 5
Use the fact that the time derivative of the logarithm of a variable equals its growth rate to show
that
a) The growth rate of the product of two variables, X (t )Y (t ) , equals the sum of their growth
rates.
b) The growth rate of the ratio of two variables, X (t ) / Y (t ) , equals the difference of their growth
rates.
c) If Z (t ) X (t ) , then Z (t ) / Z (t ) X (t ) / X (t ).
Question 6*
a) The exponential function ex is the inverse of the logarithmic function ln x . Illustrate with a
graph.
b) Does every function have an inverse? 2

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