Approximating functions with Taylor polynomials and error bounds

Approximating functions with Taylor polynomials and error bounds

Basic concepts: Higher order derivatives, Introduction to infinite series, Functions expressed as power series, Taylor and maclaurin series,
Related concepts: Linear approximation,

Lessons

To approximate a function with a Taylor Polynomial of n'th degree centred around a=0a=0, use
f(x)≈f(a)+f′(a)(x−a)+f"(a)(x−a)22!+⋯+fn(a)(x−a)2n!f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}

where Pn(x)=f(a)+f′(a)(x−a)+f"(a)(x−a)22!+⋯+fn(a)(x−a)2n!P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!} is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as
Rn(x)=fn+1(z)(x−a)n+1(n+1)!R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,
f(x)=Pn(x)+Rn(x)f(x)=P_n(x)+R_n(x)
  • Introduction
    Approximating Functions with Taylor Polynomials and Error Bounds

    i) Taylor Polynomials and the Error Term

  • 1.
    Approximate ln 2 using the 3'rd degree Taylor Polynomial. Find the error term.
  • 2.
    Find the 4th degree Taylor Polynomial centred around a=0a=0 of f(x)=exf(x)=e^x. Then approximate e2e^2.
  • 3.
    Find the 2nd degree Taylor Polynomial centred around a=1a=1 of f(x)=(x+1)f(x)=\sqrt{(x+1)} and the error term where x∈[0,2]x \in [0,2].
  • 4.
    Show that f(x)=exf(x)=e^x can be represented as a Taylor series at a=0a=0.
  • 5.
    Show that f(x)=cos⁡?xf(x)= \cos ?x can be represented as a Taylor series at a=0a=0.
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8.
Sequence and Series
8.1
Introduction to sequences
8.2
Introduction to infinite series
8.3
Convergence and divergence of normal infinite series
8.4
Convergence and divergence of geometric series
8.5
Divergence of harmonic series
8.6
P Series
8.7
Alternating series test
8.8
Divergence test
8.9
Comparison and limit comparison test
8.10
Integral test
8.11
Ratio test
8.12
Absolute and conditional convergence
8.13
Radius and interval of convergence with power series
8.14
Functions expressed as power series
8.15
Taylor and maclaurin series
8.16
Approximating functions with Taylor polynomials and error bounds
AP Calculus BC