Find the first 4 terms in the Taylor series for $( x -1) e^x$.
Answer (1)
Find the derivatives of f ( x ) = ( x − 1 ) e x , which are f ′ ( x ) = x e x , f ′′ ( x ) = ( x + 1 ) e x , and f ′′′ ( x ) = ( x + 2 ) e x .
Evaluate the derivatives at x = 0 : f ( 0 ) = − 1 , f ′ ( 0 ) = 0 , f ′′ ( 0 ) = 1 , and f ′′′ ( 0 ) = 2 .
Use the Taylor series formula: f ( x ) = f ( 0 ) + f ′ ( 0 ) x + 2 ! f ′′ ( 0 ) x 2 + 3 ! f ′′′ ( 0 ) x 3 + ...
Substitute the values to get the first four terms: − 1 + 2 x 2 + 3 x 3
Explanation
Problem Setup We are asked to find the first four terms of the Taylor series for the function f ( x ) = ( x − 1 ) e x around x = 0 . The Taylor series expansion of a function f ( x ) around x = 0 is given by: f ( x ) = f ( 0 ) + f ′ ( 0 ) x + 2 ! f ′′ ( 0 ) x 2 + 3 ! f ′′′ ( 0 ) x 3 + ... We need to find the first four derivatives of f ( x ) and evaluate them at x = 0 .
Finding Derivatives First, let's find the derivatives of f ( x ) = ( x − 1 ) e x :
f ( x ) = ( x − 1 ) e x
f ′ ( x ) = e x + ( x − 1 ) e x = x e x
f ′′ ( x ) = e x + x e x = ( x + 1 ) e x
f ′′′ ( x ) = e x + ( x + 1 ) e x = ( x + 2 ) e x
f ( 4 ) ( x ) = e x + ( x + 2 ) e x = ( x + 3 ) e x
Evaluating Derivatives at x=0 Now, let's evaluate the derivatives at x = 0 :
f ( 0 ) = ( 0 − 1 ) e 0 = − 1
f ′ ( 0 ) = 0 ⋅ e 0 = 0
f ′′ ( 0 ) = ( 0 + 1 ) e 0 = 1
f ′′′ ( 0 ) = ( 0 + 2 ) e 0 = 2
Writing the Taylor Series Now, we can write the Taylor series expansion of f ( x ) around x = 0 :
f ( x ) = f ( 0 ) + f ′ ( 0 ) x + 2 ! f ′′ ( 0 ) x 2 + 3 ! f ′′′ ( 0 ) x 3 + ...
f ( x ) = − 1 + 0 ⋅ x + 2 ! 1 x 2 + 3 ! 2 x 3 + ...
f ( x ) = − 1 + 2 1 x 2 + 3 1 x 3 + ...
Identifying the First Four Terms The first four terms of the Taylor series are:
− 1 + 0 x + 2 1 x 2 + 3 1 x 3
So, the first four terms are − 1 , 0 , 2 1 x 2 , and 3 1 x 3 .
Final Answer Therefore, the first four terms in the Taylor series for ( x − 1 ) e x are: − 1 + 0 x + 2 1 x 2 + 3 1 x 3 We can write this as: − 1 + 2 x 2 + 3 x 3
Examples
Taylor series are incredibly useful in physics and engineering. For instance, when analyzing the motion of a pendulum, the equation involves trigonometric functions like sine. For small angles, we can approximate sin ( x ) using its Taylor series expansion, simplifying the equations of motion and making the analysis much easier. This allows engineers to quickly estimate the pendulum's behavior without needing to solve complex equations.
