# 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 at the point .

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Q2:

Find the Taylor polynomials of the fourth degree approximating the function at the point .

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Q3:

Find the Taylor polynomials of the third degree approximating the function at the point .

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Q4:

Find the Taylor polynomials of the fourth degree approximating the function at the point .

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Q5:

Find the Taylor polynomials of degree two approximating the function at the point .

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Q6:

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

Suppose is twice differentiable at . Let be the polynomial satisfying , , and . In terms of , what are the coefficients , , and ?

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Is always a quadratic polynomial? Why?

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

What is when at ? Give your coefficients as fractions.

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

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

Q7:

Find the third-degree Taylor polynomial of the function at the point .

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Q8:

Use the second-degree Taylor polynomial to approximate the function at the point .

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Q9:

Find the third-degree Taylor polynomial of the function at the point .

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Q10:

Find the fourth-degree Taylor polynomial of the function at the point .

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Q11:

Estimate the function with a third-degree Taylor polynomial at .

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Q12:

Estimate the function with a third-degree Taylor polynomial at .

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Q13:

Estimate the function with a third-degree Taylor polynomial at .

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Q14:

Estimate the function with a third-degree Taylor polynomial at .

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Q15:

Estimate the function with a third-degree Taylor polynomial at .

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