Worksheet: Taylor Polynomials Approximation to a Function

In this worksheet, we will practice finding Taylor/Maclaurin polynomials and using them to approximate a function.

Q1:

Find the Taylor polynomials of the third degree approximating the function 𝑓(𝑥)=𝑥 at the point 𝑎=9.

  • A3+16(𝑥9)1108(𝑥9)+1648(𝑥9)
  • B3+16(𝑥9)1216(𝑥9)+11,944(𝑥9)
  • C3+16(𝑥9)+1216(𝑥9)+13,888(𝑥9)
  • D3+16(𝑥9)1108(𝑥9)1648(𝑥9)
  • E3+16(𝑥9)1216(𝑥9)+13,888(𝑥9)

Q2:

Find the Taylor polynomials of the fourth degree approximating the function 𝑓(𝑥)=𝑒 at the point 𝑎=3.

  • A𝑒+𝑒(𝑥3)+23𝑒(𝑥3)+13𝑒(𝑥3)+215𝑒(𝑥3)
  • B𝑒+2𝑒(𝑥3)+2𝑒(𝑥3)+83𝑒(𝑥3)+4𝑒(𝑥3)
  • C𝑒+2𝑒(𝑥3)+4𝑒(𝑥3)+8𝑒(𝑥3)+16𝑒(𝑥3)
  • D𝑒+𝑒(𝑥3)+𝑒(𝑥3)+𝑒(𝑥3)+𝑒(𝑥3)
  • E𝑒+2𝑒(𝑥3)+2𝑒(𝑥3)+43𝑒(𝑥3)+23𝑒(𝑥3)

Q3:

Find the Taylor polynomials of the third degree approximating the function 𝑓(𝑥)=12𝑥 at the point 𝑎=12.

  • A14+18𝑥12116𝑥12+124𝑥12
  • B1418𝑥12116𝑥12124𝑥12
  • C1+2𝑥12+4𝑥12+8𝑥12
  • D12𝑥124𝑥128𝑥12
  • E12𝑥12+4𝑥128𝑥12

Q4:

Find the Taylor polynomials of the fourth degree approximating the function 𝑓(𝑥)=3𝑥cos at the point 𝑎=𝜋3.

  • A1+92𝑥𝜋3+278𝑥𝜋3
  • B1+92𝑥𝜋3278𝑥𝜋3
  • C1+92𝑥𝜋3278𝑥𝜋3
  • D1+92𝑥𝜋3278𝑥𝜋3
  • E1+92𝑥𝜋3+278𝑥𝜋3

Q5:

Find the Taylor polynomials of degree two approximating the function 𝑓(𝑥)=𝑥+2𝑥3 at the point 𝑎=2.

  • A9+8(𝑥2)+32(𝑥2)
  • B9+7(𝑥2)+6(𝑥2)
  • C9+14(𝑥2)+6(𝑥2)
  • D9+14(𝑥2)+12(𝑥2)
  • E9+8(𝑥2)+3(𝑥2)

Q6:

The tangent line gives a linear approximation to a function near a point. We consider higher order polynomials.

Suppose 𝑓 is twice differentiable at 𝑥=𝑎. Let 𝑔(𝑥)=𝐴+𝐵(𝑥𝑎)+𝐶(𝑥𝑎) be the polynomial satisfying 𝑔(𝑎)=𝑓(𝑎), 𝑔(𝑎)=𝑓(𝑎), and 𝑔(𝑎)=𝑓(𝑎). In terms of 𝑓, what are the coefficients 𝐴, 𝐵, and 𝐶?

  • A𝑓(𝑎),𝑓(𝑎),𝑓(𝑎)
  • B𝑓(𝑎)𝑓(𝑎),𝑓(𝑎),𝑓(𝑎)
  • C𝑓(𝑎),𝑓(𝑎),𝑓(𝑎)
  • D𝑓(𝑎),𝑓(𝑎),𝑓(𝑎)2
  • E𝑓(𝑎)𝑓(𝑎),𝑓(𝑎),𝑓(𝑎)

Is 𝑔 always a quadratic polynomial? Why?

  • Ano, because the point 𝑎 may be an inflection point of 𝑓
  • Byes, because we have coefficients up to degree 2

What is 𝑔(𝑥) when 𝑓(𝑥)=𝑥 at 𝑥=8? Give your coefficients as fractions.

  • A1144+1288(𝑥8)+1576(𝑥8)
  • B2+2(𝑥8)+112(𝑥8)
  • C2+112(𝑥8)1288(𝑥8)
  • D2+112(𝑥8)1144(𝑥8)
  • E16+112(𝑥8)+112(𝑥8)

Using the function 𝑥 at 𝑥=8, find the tangent line approximation of the cube root of 7 to 5 decimal places.

Using the function 𝑥 at 𝑥=8, determine the quadratic approximation of the cube root of 7 to 5 decimal places.

Q7:

Find the third-degree Taylor polynomial of the function 𝑓(𝑥)=1𝑥 at the point 𝑎=2.

  • A1414(𝑥2)+316(𝑥2)18(𝑥2)
  • B14+14(𝑥2)+38(𝑥2)+34(𝑥2)
  • C1418(𝑥2)+116(𝑥2)132(𝑥2)
  • D14+14(𝑥2)+316(𝑥2)+18(𝑥2)
  • E1414(𝑥2)+38(𝑥2)34(𝑥2)

Q8:

Use the second-degree Taylor polynomial to approximate the function 𝑓(𝑥)=2+𝑥+𝑥 at the point 𝑎=1.

  • A4+3(𝑥1)+(𝑥1)
  • B4+32(1𝑥)+23(1𝑥)
  • C4+32(𝑥1)+23(𝑥1)
  • D4+3(1𝑥)+(1𝑥)
  • E4+3(𝑥1)+2(𝑥1)

Q9:

Find the third-degree Taylor polynomial of the function 𝑓(𝑥)=2𝑥ln at the point 𝑎=1.

  • Aln2+(𝑥1)12(𝑥1)+13(𝑥1)
  • Bln2+12(𝑥1)13(𝑥1)+14(𝑥1)
  • Cln2+(𝑥1)(𝑥1)+(𝑥1)
  • Dln2+(𝑥1)+12(𝑥1)+13(𝑥1)
  • Eln2+(𝑥1)+(𝑥1)+(𝑥1)

Q10:

Find the fourth-degree Taylor polynomial of the function 𝑓(𝑥)=𝑥sin at the point 𝑎=𝜋2.

  • A1+𝑥𝜋218𝑥𝜋2
  • B1𝑥𝜋2+18𝑥𝜋2
  • C112𝑥𝜋2+124𝑥𝜋2
  • D112𝑥𝜋2+124𝑥𝜋2
  • E1+12𝑥𝜋2124𝑥𝜋2

Q11:

Estimate the function 𝑓(𝑥)=2(3𝑥)tan with a third-degree Taylor polynomial at 𝑥=𝜋.

  • A6(𝑥𝜋)18(𝑥𝜋)
  • B6(𝑥𝜋)+18(𝑥𝜋)
  • C6(𝑥𝜋)
  • D2(𝑥𝜋)+23(𝑥𝜋)
  • E6(𝑥𝜋)18(𝑥𝜋)

Q12:

Estimate the function 𝑓(𝑥)=2𝑥(3𝑥)ln with a third-degree Taylor polynomial at 𝑥=1.

  • A23+(23+2)(𝑥1)+(𝑥1)+13(𝑥1)lnln
  • B23+2(𝑥1)(𝑥1)+23(𝑥1)ln
  • C23+(23+2)(𝑥1)+2(𝑥1)2(𝑥1)lnln
  • D23+(23+2)(𝑥1)+(𝑥1)13(𝑥1)lnln
  • E23+23+23(𝑥1)+13(𝑥1)19(𝑥1)lnln

Q13:

Estimate the function 𝑓(𝑥)=2(3𝑥)sin with a third-degree Taylor polynomial at 𝑥=𝜋.

  • A2(𝑥𝜋)+13(𝑥𝜋)
  • B6(𝑥𝜋)+54(𝑥𝜋)
  • C15(𝑥𝜋)1252(𝑥𝜋)
  • D6(𝑥𝜋)9(𝑥𝜋)
  • E6(𝑥𝜋)+9(𝑥𝜋)

Q14:

Estimate the function 𝑓(𝑥)=23𝑥 with a third-degree Taylor polynomial at 𝑥=1.

  • A23333(𝑥1)+32(𝑥1)534(𝑥1)
  • B233+33(𝑥1)+34(𝑥1)+5324(𝑥1)
  • C23333(𝑥1)+34(𝑥1)5324(𝑥1)
  • D233+33(𝑥1)312(𝑥1)+324(𝑥1)
  • E233+33(𝑥1)34(𝑥1)+5324(𝑥1)

Q15:

Estimate the function 𝑓(𝑥)=2𝑒 with a third-degree Taylor polynomial at 𝑥=1.

  • A2𝑒+6𝑒(𝑥1)+9𝑒(𝑥1)+9𝑒(𝑥1)
  • B2𝑒+6𝑒(𝑥1)9𝑒(𝑥1)+9𝑒(𝑥1)
  • C2𝑒+2𝑒(𝑥1)+𝑒(𝑥1)+13𝑒(𝑥1)
  • D2𝑒6𝑒(𝑥1)+9𝑒(𝑥1)9𝑒(𝑥1)
  • E2𝑒6𝑒(𝑥1)+18𝑒(𝑥1)54𝑒(𝑥1)

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