# Worksheet: Linear Approximation

In this worksheet, we will practice using derivatives to find the equation of the line that approximates the function near a certain value and using differentials to approximate the change in the function.

**Q5: **

Find the linear approximation of the function at .

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- E

**Q6: **

Find the linear approximation of the function at .

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**Q8: **

Find the linear approximation of the function at .

**Q9: **

Find the linear approximation of the function at .

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- B0
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**Q11: **

Find the linear approximation of the function at .

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**Q12: **

Find the linear approximation of the function at .

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**Q14: **

By finding the linear approximation of the function at a suitable value of , estimate the value of .

**Q16: **

By finding the linear approximation of the function at a suitable value of , estimate the value of .

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**Q17: **

We will explore why we can call the tangent line approximation the “best” local linearization.

What is the tangent line approximation at for the function ?

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Suppose is used as a local linearization at of . Write the expression for the error .

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Determine the value of for which

- A3.8
- B1
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- E0

Suppose that is a function that is differentiable at . Using the local linearization at given by , determine

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**Q18: **

By finding the linear approximation of the function at a suitable value of , estimate the value of .

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- B2
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**Q20: **

Use a local linearization near to estimate to 3 decimal places, where .

**Q21: **

By finding the linear approximation of the function at a suitable value of , estimate the value of .

**Q23: **

The first condition for a linear function to be an approximation to near is that . So for some constant . The error in using instead of at a point is the function .

The tangent line approximation to near is the linear approximation . What is the tangent line approximation to near ?

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What is the error in using the tangent line approximation to near at the point 0.1? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to near at the point 0.01? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to near at the point 0.001? Give your answer to 5 decimal places.

**Q24: **

The curve of equation is concave up everywhere. The line is above when but then must quickly cross at some near 0.

What equation can we use to find ?

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Find an estimate of using the tangent line approximation of at in the equation above. Give your answer to 3 decimal places.

**Q25: **

In the figure, which is the graph of the tangent line approximation of near ?

- A(a)
- B(b)
- C(d)
- Dnone of the lines shown
- E(c)