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Taylor Series, Logarithmic and Exponential Function
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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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