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Chapter exploration

Maclaurin series and approximation

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For e2xe^{2x}, compare the quadratic polynomial P2=1+2x+2x2P_2=1+2x+2x^2 with P4=P2+4x3/3+2x4/3P_4=P_2+4x^3/3+2x^4/3. Predict which is closer at x=0.1x=0.1, then compare both errors using a calculator or graph.

Repeat at x=0.5x=0.5 and x=2x=2 using the same scale for all three curves at each input.

Explain why convergence of the exponential series for all real x does not mean a fixed low-degree polynomial is accurate at every input. The numerical comparisons assess these inputs, without proving a general error bound.

Related interactive example

Explore this related example, then return to the prediction task above.

Adding terms to 2(1 − 3x/4)^(1/2)

Compare the function with a finite polynomial. Change the highest included power and test x, then use the error plot to see how the approximation changes at that x.

Highest included powerxⁿ, n = 2
Test value xx = 0.25
Reset the controls
Function
1.80278
P₂(x)
1.80371
Error
0.00094

Inside the stated open interval. A finite truncation can still have a noticeable error.

Teal marks the function, amber marks the polynomial, and the red segment joins their values at the selected x.

Does keeping more powers help at this x?

Highest included power n runs horizontally; signed error Pₙ(x) − f(x) runs vertically. Zero means agreement at this x. A missing power leaves the polynomial unchanged; individual steps need not improve the error.

  • Function
  • Series truncated at the chosen power
  • Stated open interval. Endpoint convergence requires a separate check
Key idea

Inside −43<x<43-\frac43<x<\frac43 extra terms improve the approximation; the displayed truncated polynomial approximates the function there.

Ideas to try
  1. Slide xx towards 43\frac43: does the polynomial still approach the curve?
  2. Move xx beyond 43\frac43 and see what happens to the readouts.
Open the full interactive problem →
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