**Q1: **

Find the linear approximation of the function at .

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**Q2: **

Find the linear approximation of the function at .

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**Q3: **

What is the tangent line approximation of near ?

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**Q4: **

Find the linear approximation of the function at .

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**Q5: **

Find the linear approximation of the function at .

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**Q6: **

Find the linear approximation of the function at .

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**Q7: **

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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- E0

**Q10: **

Find the linear approximation of the function at .

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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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**Q13: **

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

- A15.984
- B15.992
- C16.032
- D15.968
- E16.016

**Q14: **

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

**Q15: **

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 βbestβ 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

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- B3.8
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- D0
- E1

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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**Q19: **

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

- A0
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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 .

**Q22: **

Find the local linearization of near .

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**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 ?

- A(b)
- B(a)
- C(c)
- D(d)
- Enone of the lines shown

**Q26: **

Suppose that is any linear approximation to the function near . The error in using instead of at a point is given by . What is the error function in using the tangent line approximation for near ?

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**Q27: **

Let be the tangent line approximation of the function at , so that is its error function. Since the function as , we can locally linearize it by for some constant at . This means that or . Estimate to 3 decimal places by examining a suitable sequence of values.

**Q28: **

Suppose that we call the linear function a good approximation to near provided that

What does this tell us about ?

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- B
- C There is nothing we can deduce about the value of .
- D

So, we can rewrite . Suppose that we have another good linear approximation to . How is related to ?

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Suppose and are differentiable at . Then their tangent line approximations and are both good approximations near . Use these two to find a good linear approximation for the product function near in the form .

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What can you conclude about the derivative ?

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**Q29: **

Use the approximation , which is valid for small values of , to estimate .

- A0.96
- B0.99
- C1.03
- D1.01
- E1.02

**Q30: **

We want to find cube roots using the cube function .

The cube root of 7 can be found as the zero of , which is just . What is the local linearization of at ?

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- C1
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By finding the intersection with the -axis, what does the tangent line linearization give as an estimate of ? Answer with a fraction. We will call this value .

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Using the value above, what is the tangent line linearization of at ? Give an exact answer.

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Using this new tangent line, give a second estimate of the cube root accurate to five decimal places.

- A1.91294
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- E 1.92039

**Q31: **

By factoring the error function in using the tangent line approximation to near , or otherwise, find the largest so that if then this error is less than 0.01. Give your to 3 decimal places.

**Q32: **

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?

- A no, because the point may be an inflection point of
- B yes, 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.

**Q33: **

We consider the local linearization of the natural log function near .

Find the local linearization of near .

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Knowing now that , where is the error function, what is ?