riaology
Unit 4 · Lesson 9 of 9

Match more derivatives to improve a local approximation.

An exponential response can be approximated near zero by a line, a quadratic, or a higher-degree polynomial. Compare error as you change degree.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its Taylor-polynomial value.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase input x? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelPN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

Input x versus exp(x). Exact exp(x), Taylor polynomial. Selected Taylor-polynomial value: 2.66667 . Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.-0.333-23.52807.3892exp(x)Input x
Exact exp(x)Taylor polynomial · dashed

Taylor-polynomial valueP3(1)=2.66667,ex−PN=0.0516152P_3(1)=2.66667,\quad e^x-P_N=0.0516152

Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.

Assumptions, units, and limits

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

PN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}
02

Keep the reasoning visible

PN(x)=∑n=0Nf(n)(a)n!(x−a)nP_N(x)=\sum_{n=0}^N\frac{f^{(n)}(a)}{n!}(x-a)^n

Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.

03

What makes this model useful?

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

Convergence of a formal Taylor series does not in general prove equality with its generating function. A remainder argument supplies that connection.

A worked example

Calculate the starting scenario

  1. Use input x = 1 ; polynomial degree n = 3 .
  2. P3(1)=2.66667,ex−PN=0.0516152P_3(1)=2.66667,\quad e^x-P_N=0.0516152
  3. Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x. The checked quantity is Taylor-polynomial value.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the Taylor-polynomial value using input x 1 , polynomial degree n 3 . State the meaning of its sign and units.

Checked quantity: Taylor-polynomial value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

Explore the visual solution
Read the modelPN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

Input x versus exp(x). Exact exp(x), Taylor polynomial. Selected Taylor-polynomial value: 2.66667 . Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.-0.333-23.52807.3892exp(x)Input x
Exact exp(x)Taylor polynomial · dashed

Taylor-polynomial valueP3(1)=2.66667,ex−PN=0.0516152P_3(1)=2.66667,\quad e^x-P_N=0.0516152

Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.

Assumptions, units, and limits

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

PN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}
Calculate and explain

Apply the idea to a new scenario

Use degree 4 at x=1.5x=1.5 to approximate exp(x). Find the polynomial value and signed exact-minus-approximation error.

Checked quantity: Taylor-polynomial value · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

Explore the visual solution
Read the modelPN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

Input x versus exp(x). Exact exp(x), Taylor polynomial. Selected Taylor-polynomial value: 4.39844 . Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.0-23.69507.3892exp(x)Input x
Exact exp(x)Taylor polynomial · dashed

Taylor-polynomial valueP4(1.5)=4.39844,ex−PN=0.0832516P_4(1.5)=4.39844,\quad e^x-P_N=0.0832516

Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.

Assumptions, units, and limits

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

PN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the modelPN(x)=∑n=0Nxnn!,ex=∑n=0∞xnn!P_N(x)=\sum_{n=0}^N\frac{x^n}{n!},\quad e^x=\sum_{n=0}^\infty\frac{x^n}{n!}

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

Input x versus exp(x). Exact exp(x), Taylor polynomial. Selected Taylor-polynomial value: 2.66667 . Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.-0.333-23.52807.3892exp(x)Input x
Exact exp(x)Taylor polynomial · dashed

Taylor-polynomial valueP3(1)=2.66667,ex−PN=0.0516152P_3(1)=2.66667,\quad e^x-P_N=0.0516152

Matching successive derivatives at the center creates progressively higher-order local approximations. The infinite series here converges for every real x.

Assumptions, units, and limits

Maclaurin expansion of exp(x). Finite polynomial accuracy depends on degree and distance from the center.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Taylor coefficients are derivatives at the center divided by factorials. A finite Taylor polynomial is an approximation; an infinite Taylor series equals the function where the remainder tends to zero. For exp(x), the series converges to the function everywhere.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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