Problem 4. (1 point)
The linearization at $a=0$ to $\sin (3 x)$ is $A+B x$. Compute $A$ and $B$.
$\begin{array}{l}
A= \\
B=
\end{array}$
Answer(s) submitted:
Answer (1)
Find the value of the function at a = 0 : f ( 0 ) = sin ( 3 ⋅ 0 ) = 0 , so A = 0 .
Find the derivative of the function: f ′ ( x ) = 3 cos ( 3 x ) .
Evaluate the derivative at a = 0 : f ′ ( 0 ) = 3 cos ( 0 ) = 3 , so B = 3 .
The linearization is A + B x , thus A = 0 and B = 3 .
Explanation
Problem Analysis We are asked to find the linearization of sin ( 3 x ) at a = 0 and express it in the form A + B x . This means we need to find the values of A and B .
Linearization Formula The linearization of a function f ( x ) at a point x = a is given by the formula: L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) In our case, f ( x ) = sin ( 3 x ) and a = 0 .
Finding A First, we need to find the value of the function at a = 0 :
f ( 0 ) = sin ( 3 ⋅ 0 ) = sin ( 0 ) = 0 So, A = f ( 0 ) = 0 .
Finding B Next, we need to find the derivative of the function f ( x ) = sin ( 3 x ) . Using the chain rule, we get: f ′ ( x ) = 3 cos ( 3 x ) Now, we evaluate the derivative at a = 0 :
f ′ ( 0 ) = 3 cos ( 3 ⋅ 0 ) = 3 cos ( 0 ) = 3 ⋅ 1 = 3 So, B = f ′ ( 0 ) = 3 .
The Linearization Therefore, the linearization of sin ( 3 x ) at a = 0 is: L ( x ) = 0 + 3 ( x − 0 ) = 3 x Comparing this with A + B x , we have A = 0 and B = 3 .
Final Answer Thus, the values of A and B are A = 0 and B = 3 .
A = 0 B = 3
Examples
Linearization is used in physics to approximate the behavior of a system near an equilibrium point. For example, the motion of a pendulum can be approximated as simple harmonic motion for small angles using linearization. This simplifies the analysis and allows for easier calculations of the pendulum's period and frequency. Understanding linearization helps in modeling and predicting the behavior of complex systems in various fields of science and engineering.
