12..1. Taylor Series Expansion – The Basic Idea

Try these first

1. 1.

   What is a Taylor Series?

   Hint

   Think of how we can express functions as sums.

2. 2.

   What is the formula for the first-order derivative?

   Hint

   Consider it as the slope of the tangent line.

3. 3.

   What is the primary function of the Taylor Series method?

   • To find exact solutions
   • To approximate solutions
   • To manipulate equations

   Hint

   Think about why we use numerical methods.

4. 4.

   True or False: The Taylor Series can be applied to any function.

   • True
   • False

   Hint

   Consider the function types appropriate for Taylor series.

5. 5.

   Using the Taylor Series method, calculate y(0.2) for the ODE dy/dx = x^2 - y, with the initial condition y(0) = 0 and a step size of 0.2.

   Hint

   Ensure you're computing each derivative accurately before substituting.

6. 6.

   Derive the Taylor Series for e^x up to the third degree and use it to approximate e^0.1.

   Hint

   Use the Taylor series formula for e^x, which is well-known!