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 .

  • A 3 + 1 6 ( 𝑥 9 ) 1 1 0 8 ( 𝑥 9 ) + 1 6 4 8 ( 𝑥 9 )
  • B 3 + 1 6 ( 𝑥 9 ) 1 2 1 6 ( 𝑥 9 ) + 1 1 , 9 4 4 ( 𝑥 9 )
  • C 3 + 1 6 ( 𝑥 9 ) + 1 2 1 6 ( 𝑥 9 ) + 1 3 , 8 8 8 ( 𝑥 9 )
  • D 3 + 1 6 ( 𝑥 9 ) 1 2 1 6 ( 𝑥 9 ) + 1 3 , 8 8 8 ( 𝑥 9 )
  • E 3 + 1 6 ( 𝑥 9 ) 1 1 0 8 ( 𝑥 9 ) 1 6 4 8 ( 𝑥 9 )

Q2:

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

  • A 𝑒 + 2 𝑒 ( 𝑥 3 ) + 4 𝑒 ( 𝑥 3 ) + 8 𝑒 ( 𝑥 3 ) + 1 6 𝑒 ( 𝑥 3 )
  • B 𝑒 + 2 𝑒 ( 𝑥 3 ) + 2 𝑒 ( 𝑥 3 ) + 4 3 𝑒 ( 𝑥 3 ) + 2 3 𝑒 ( 𝑥 3 )
  • C 𝑒 + 𝑒 ( 𝑥 3 ) + 𝑒 ( 𝑥 3 ) + 𝑒 ( 𝑥 3 ) + 𝑒 ( 𝑥 3 )
  • D 𝑒 + 2 𝑒 ( 𝑥 3 ) + 2 𝑒 ( 𝑥 3 ) + 8 3 𝑒 ( 𝑥 3 ) + 4 𝑒 ( 𝑥 3 )
  • E 𝑒 + 𝑒 ( 𝑥 3 ) + 2 3 𝑒 ( 𝑥 3 ) + 1 3 𝑒 ( 𝑥 3 ) + 2 1 5 𝑒 ( 𝑥 3 )

Q3:

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

  • A 1 2 𝑥 1 2 4 𝑥 1 2 8 𝑥 1 2
  • B 1 4 1 8 𝑥 1 2 1 1 6 𝑥 1 2 1 2 4 𝑥 1 2
  • C 1 4 + 1 8 𝑥 1 2 1 1 6 𝑥 1 2 + 1 2 4 𝑥 1 2
  • D 1 2 𝑥 1 2 + 4 𝑥 1 2 8 𝑥 1 2
  • E 1 + 2 𝑥 1 2 + 4 𝑥 1 2 + 8 𝑥 1 2

Q4:

Find the Taylor polynomials of the fourth degree approximating the function 𝑓 ( 𝑥 ) = 3 𝑥 c o s at the point 𝑎 = 𝜋 3 .

  • A 1 + 9 2 𝑥 𝜋 3 2 7 8 𝑥 𝜋 3
  • B 1 + 9 2 𝑥 𝜋 3 2 7 8 𝑥 𝜋 3
  • C 1 + 9 2 𝑥 𝜋 3 2 7 8 𝑥 𝜋 3
  • D 1 + 9 2 𝑥 𝜋 3 + 2 7 8 𝑥 𝜋 3
  • E 1 + 9 2 𝑥 𝜋 3 + 2 7 8 𝑥 𝜋 3

Q5:

Find the Taylor polynomials of degree two approximating the function 𝑓 ( 𝑥 ) = 𝑥 + 2 𝑥 3 at the point 𝑎 = 2 .

  • A 9 + 1 4 ( 𝑥 2 ) + 6 ( 𝑥 2 )
  • B 9 + 1 4 ( 𝑥 2 ) + 1 2 ( 𝑥 2 )
  • C 9 + 8 ( 𝑥 2 ) + 3 2 ( 𝑥 2 )
  • D 9 + 7 ( 𝑥 2 ) + 6 ( 𝑥 2 )
  • E 9 + 8 ( 𝑥 2 ) + 3 ( 𝑥 2 )

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 ( 𝑥 𝑎 ) + 𝐶 ( 𝑥 𝑎 ) 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 ?

  • A 𝑓 ( 𝑎 ) 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 )
  • B 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 )
  • C 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 )
  • D 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) 2
  • E 𝑓 ( 𝑎 ) 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 ) , 𝑓 ( 𝑎 )

Is 𝑔 always a quadratic polynomial? Why?

  • A yes, because we have coefficients up to degree 2
  • B no, because the point 𝑎 may be an inflection point of 𝑓

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

  • A 2 + 1 1 2 ( 𝑥 8 ) 1 2 8 8 ( 𝑥 8 )
  • B 2 + 1 1 2 ( 𝑥 8 ) 1 1 4 4 ( 𝑥 8 )
  • C 1 6 + 1 1 2 ( 𝑥 8 ) + 1 1 2 ( 𝑥 8 )
  • D 2 + 2 ( 𝑥 8 ) + 1 1 2 ( 𝑥 8 )
  • E 1 1 4 4 + 1 2 8 8 ( 𝑥 8 ) + 1 5 7 6 ( 𝑥 8 )

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 ) = 1 𝑥 at the point 𝑎 = 2 .

  • A 1 4 1 4 ( 𝑥 2 ) + 3 1 6 ( 𝑥 2 ) 1 8 ( 𝑥 2 )
  • B 1 4 1 4 ( 𝑥 2 ) + 3 8 ( 𝑥 2 ) 3 4 ( 𝑥 2 )
  • C 1 4 + 1 4 ( 𝑥 2 ) + 3 1 6 ( 𝑥 2 ) + 1 8 ( 𝑥 2 )
  • D 1 4 + 1 4 ( 𝑥 2 ) + 3 8 ( 𝑥 2 ) + 3 4 ( 𝑥 2 )
  • E 1 4 1 8 ( 𝑥 2 ) + 1 1 6 ( 𝑥 2 ) 1 3 2 ( 𝑥 2 )

Q8:

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

  • A 4 + 3 ( 1 𝑥 ) + ( 1 𝑥 )
  • B 4 + 3 ( 𝑥 1 ) + ( 𝑥 1 )
  • C 4 + 3 2 ( 𝑥 1 ) + 2 3 ( 𝑥 1 )
  • D 4 + 3 2 ( 1 𝑥 ) + 2 3 ( 1 𝑥 )
  • E 4 + 3 ( 𝑥 1 ) + 2 ( 𝑥 1 )

Q9:

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

  • A l n 2 + ( 𝑥 1 ) ( 𝑥 1 ) + ( 𝑥 1 )
  • B l n 2 + ( 𝑥 1 ) + ( 𝑥 1 ) + ( 𝑥 1 )
  • C l n 2 + ( 𝑥 1 ) 1 2 ( 𝑥 1 ) + 1 3 ( 𝑥 1 )
  • D l n 2 + ( 𝑥 1 ) + 1 2 ( 𝑥 1 ) + 1 3 ( 𝑥 1 )
  • E l n 2 + 1 2 ( 𝑥 1 ) 1 3 ( 𝑥 1 ) + 1 4 ( 𝑥 1 )

Q10:

Find the fourth-degree Taylor polynomial of the function 𝑓 ( 𝑥 ) = 𝑥 s i n at the point 𝑎 = 𝜋 2 .

  • A 1 1 2 𝑥 𝜋 2 + 1 2 4 𝑥 𝜋 2
  • B 1 1 2 𝑥 𝜋 2 + 1 2 4 𝑥 𝜋 2
  • C 1 + 1 2 𝑥 𝜋 2 1 2 4 𝑥 𝜋 2
  • D 1 + 𝑥 𝜋 2 1 8 𝑥 𝜋 2
  • E 1 𝑥 𝜋 2 + 1 8 𝑥 𝜋 2

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