Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
Example 3.3.1
For fx=ex at x=0, obtain Taylor polynomials of degree 1, 2, and 3.
Graphically compare these polynomials on −4,4.
Use R3x to estimate the largest difference between f and P3 on −1,1.
Find the actual value of the largest difference between f and P3 on −1,1.
Solution
Initialize and define f
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Type fx=ⅇx, being sure to use the exponential "e" (palette or Command Completion).
Context Panel: Assign Function
fx=ⅇx→assign as functionf
Invoke the tutor
Type fx Context Panel: Evaluate and Display Inline
Context Panel: Tutors≻Taylor Approximation
fx = ⅇx→Taylor approximation tutor
Figure 3.3.1(a) contains an image of the Taylor Approximation tutor adjusted for Degree = 3 (Default is 4).
Figure 3.3.1(b) provides an animation showing the first three Taylor polynomials. The Animate button in the tutor also produces an animation, but that animation shows P3 and the next four approximations.
Figure 3.3.1(a) Taylor Approximation tutor
Student:-Calculus1:-TaylorApproximation(exp(x),0,degree=3,view=[-4..4,-5.44..26],output=animation,caption="",title="",order=1..3);
Figure 3.3.1(b) Animation showing f and P1,P2,P3
Taylor Polynomials from the Context Panel
Context Panel: Series≻Series≻x Series order≻4 Select "Remove order term" (See Figure 3.3.1(c).)
The Series expansion point is the "a" in the Taylor Formula. The Series order is the degree of Rn; it is one higher than the degree of Pn. Removing the order term deletes the symbol Ox4, which stands for R4.
fx = ⅇx→series in x1+x+12x2+16x3
Figure 3.3.1(c) Context Panel for Series
Taylor Polynomials from first principles
Expression palette: Summation template Type the mathematical notation for P3x. Context Panel: Evaluate and Display Inline
∑k=03fk0k!xk = 1+x+12x2+16x3
Alternatively, compute each coefficient of P3 separately and assemble the polynomial according to the Taylor Formula.
Since fk0=1 for integer k>0, Pnx=∑k=0nxkk!.
f0 = 1
f′0 = 1
f″02! = 12
f‴03! = 16
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