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Lesson Worksheet: Taylor Polynomials Approximation to a Function Mathematics • Higher Education

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


Find the Taylor polynomials of the fourth degree approximating the function 𝑓(π‘₯)=π‘’οŠ¨ο— at the point π‘Ž=3.

  • A𝑒+𝑒(π‘₯βˆ’3)+23𝑒(π‘₯βˆ’3)+13𝑒(π‘₯βˆ’3)+215𝑒(π‘₯βˆ’3)οŠͺ
  • B𝑒+2𝑒(π‘₯βˆ’3)+2𝑒(π‘₯βˆ’3)+83𝑒(π‘₯βˆ’3)+4𝑒(π‘₯βˆ’3)οŠͺ
  • C𝑒+2𝑒(π‘₯βˆ’3)+4𝑒(π‘₯βˆ’3)+8𝑒(π‘₯βˆ’3)+16𝑒(π‘₯βˆ’3)οŠͺ
  • D𝑒+𝑒(π‘₯βˆ’3)+𝑒(π‘₯βˆ’3)+𝑒(π‘₯βˆ’3)+𝑒(π‘₯βˆ’3)οŠͺ
  • E𝑒+2𝑒(π‘₯βˆ’3)+2𝑒(π‘₯βˆ’3)+43𝑒(π‘₯βˆ’3)+23𝑒(π‘₯βˆ’3)οŠͺ


Find the Taylor polynomials of the third degree approximating the function 𝑓(π‘₯)=√π‘₯ at the point π‘Ž=9.

  • A3+16(π‘₯βˆ’9)βˆ’1108(π‘₯βˆ’9)+1648(π‘₯βˆ’9)
  • B3+16(π‘₯βˆ’9)βˆ’1216(π‘₯βˆ’9)+11,944(π‘₯βˆ’9)
  • C3+16(π‘₯βˆ’9)+1216(π‘₯βˆ’9)+13,888(π‘₯βˆ’9)
  • D3+16(π‘₯βˆ’9)βˆ’1108(π‘₯βˆ’9)βˆ’1648(π‘₯βˆ’9)
  • E3+16(π‘₯βˆ’9)βˆ’1216(π‘₯βˆ’9)+13,888(π‘₯βˆ’9)


Find the third-degree Taylor polynomial approximating the function 𝑓(π‘₯)=12π‘₯ at the point π‘Ž=12.

  • A14+18ο€Όπ‘₯βˆ’12οˆβˆ’116ο€Όπ‘₯βˆ’12+124ο€Όπ‘₯βˆ’12
  • B14βˆ’18ο€Όπ‘₯βˆ’12οˆβˆ’116ο€Όπ‘₯βˆ’12οˆβˆ’124ο€Όπ‘₯βˆ’12
  • C1+2ο€Όπ‘₯βˆ’12+4ο€Όπ‘₯βˆ’12+8ο€Όπ‘₯βˆ’12
  • D1βˆ’2ο€Όπ‘₯βˆ’12οˆβˆ’4ο€Όπ‘₯βˆ’12οˆβˆ’8ο€Όπ‘₯βˆ’12
  • E1βˆ’2ο€Όπ‘₯βˆ’12+4ο€Όπ‘₯βˆ’12οˆβˆ’8ο€Όπ‘₯βˆ’12


Find the Taylor polynomials of the fourth degree approximating the function 𝑓(π‘₯)=3π‘₯cos at the point π‘Ž=πœ‹3.

  • Aβˆ’1+92ο€»π‘₯βˆ’πœ‹3+278ο€»π‘₯βˆ’πœ‹3ο‡οŠ¨οŠͺ
  • B1+92ο€»π‘₯βˆ’πœ‹3ο‡βˆ’278ο€»π‘₯βˆ’πœ‹3ο‡οŠ©
  • Cβˆ’1+92ο€»π‘₯βˆ’πœ‹3ο‡βˆ’278ο€»π‘₯βˆ’πœ‹3ο‡οŠ¨οŠ©
  • Dβˆ’1+92ο€»π‘₯βˆ’πœ‹3ο‡βˆ’278ο€»π‘₯βˆ’πœ‹3ο‡οŠ¨οŠͺ
  • E1+92ο€»π‘₯βˆ’πœ‹3+278ο€»π‘₯βˆ’πœ‹3ο‡οŠ¨οŠͺ


Find the Taylor polynomials of degree two approximating the function 𝑓(π‘₯)=π‘₯+2π‘₯βˆ’3 at the point π‘Ž=2.

  • A9+8(π‘₯βˆ’2)+32(π‘₯βˆ’2)
  • B9+7(π‘₯βˆ’2)+6(π‘₯βˆ’2)
  • C9+14(π‘₯βˆ’2)+6(π‘₯βˆ’2)
  • D9+14(π‘₯βˆ’2)+12(π‘₯βˆ’2)
  • E9+8(π‘₯βˆ’2)+3(π‘₯βˆ’2)


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

Suppose 𝑓 is twice differentiable at π‘₯=π‘Ž. Let 𝑔(π‘₯)=𝐴+𝐡(π‘₯βˆ’π‘Ž)+𝐢(π‘₯βˆ’π‘Ž) be the polynomial satisfying 𝑔(π‘Ž)=𝑓(π‘Ž), 𝑔′(π‘Ž)=𝑓′(π‘Ž), and 𝑔′′(π‘Ž)=𝑓′′(π‘Ž). In terms of 𝑓, what are the coefficients 𝐴, 𝐡, and 𝐢?

  • A𝑓(π‘Ž),𝑓(π‘Ž),𝑓′(π‘Ž)
  • B𝑓′′(π‘Ž)𝑓(π‘Ž),𝑓′′(π‘Ž),𝑓′′(π‘Ž)
  • C𝑓(π‘Ž),𝑓′(π‘Ž),𝑓′′(π‘Ž)
  • D𝑓(π‘Ž),𝑓′(π‘Ž),𝑓′′(π‘Ž)2
  • E𝑓′(π‘Ž)𝑓(π‘Ž),𝑓′(π‘Ž),𝑓′(π‘Ž)

Is 𝑔 always a quadratic polynomial? Why?

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

What is 𝑔(π‘₯) when 𝑓(π‘₯)=√π‘₯ at π‘₯=8? Give your coefficients as fractions.

  • A1144+1288(π‘₯βˆ’8)+1576(π‘₯βˆ’8)
  • B2+2(π‘₯βˆ’8)+112(π‘₯βˆ’8)
  • C2+112(π‘₯βˆ’8)βˆ’1288(π‘₯βˆ’8)
  • D2+112(π‘₯βˆ’8)βˆ’1144(π‘₯βˆ’8)
  • E16+112(π‘₯βˆ’8)+112(π‘₯βˆ’8)

Using the function √π‘₯ at π‘₯=8, find the tangent line approximation of the cube root of 7 to 5 decimal places.

Using the function √π‘₯ at π‘₯=8, determine the quadratic approximation of the cube root of 7 to 5 decimal places.


Use the second-degree Taylor polynomial to approximate the function 𝑓(π‘₯)=2+π‘₯+π‘₯ at the point π‘Ž=1.

  • A4+3(π‘₯βˆ’1)+(π‘₯βˆ’1)
  • B4+32(1βˆ’π‘₯)+23(1βˆ’π‘₯)
  • C4+32(π‘₯βˆ’1)+23(π‘₯βˆ’1)
  • D4+3(1βˆ’π‘₯)+(1βˆ’π‘₯)
  • E4+3(π‘₯βˆ’1)+2(π‘₯βˆ’1)


Find the third-degree Taylor polynomial of the function 𝑓(π‘₯)=1π‘₯ at the point π‘Ž=2.

  • A14βˆ’14(π‘₯βˆ’2)+316(π‘₯βˆ’2)βˆ’18(π‘₯βˆ’2)
  • B14+14(π‘₯βˆ’2)+38(π‘₯βˆ’2)+34(π‘₯βˆ’2)
  • C14βˆ’18(π‘₯βˆ’2)+116(π‘₯βˆ’2)βˆ’132(π‘₯βˆ’2)
  • D14+14(π‘₯βˆ’2)+316(π‘₯βˆ’2)+18(π‘₯βˆ’2)
  • E14βˆ’14(π‘₯βˆ’2)+38(π‘₯βˆ’2)βˆ’34(π‘₯βˆ’2)


Find the third-degree Taylor polynomial of the function 𝑓(π‘₯)=2π‘₯ln at the point π‘Ž=1.

  • Aln2+(π‘₯βˆ’1)βˆ’12(π‘₯βˆ’1)+13(π‘₯βˆ’1)
  • Bln2+12(π‘₯βˆ’1)βˆ’13(π‘₯βˆ’1)+14(π‘₯βˆ’1)
  • Cln2+(π‘₯βˆ’1)βˆ’(π‘₯βˆ’1)+(π‘₯βˆ’1)
  • Dln2+(π‘₯βˆ’1)+12(π‘₯βˆ’1)+13(π‘₯βˆ’1)
  • Eln2+(π‘₯βˆ’1)+(π‘₯βˆ’1)+(π‘₯βˆ’1)


Find the fourth-degree Taylor polynomial of the function 𝑓(π‘₯)=π‘₯sin at the point π‘Ž=πœ‹2.

  • A1+ο€»π‘₯βˆ’πœ‹2ο‡βˆ’18ο€»π‘₯βˆ’πœ‹2ο‡οŠ¨οŠͺ
  • B1βˆ’ο€»π‘₯βˆ’πœ‹2+18ο€»π‘₯βˆ’πœ‹2ο‡οŠ¨οŠͺ
  • C1βˆ’12ο€»π‘₯βˆ’πœ‹2+124ο€»π‘₯βˆ’πœ‹2ο‡οŠ¨οŠͺ
  • D1βˆ’12ο€»π‘₯βˆ’πœ‹2+124ο€»π‘₯βˆ’πœ‹2ο‡οŠ¨
  • E1+12ο€»π‘₯βˆ’πœ‹2ο‡βˆ’124ο€»π‘₯βˆ’πœ‹2ο‡οŠ¨

This lesson includes 5 additional questions and 45 additional question variations for subscribers.

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