# 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.

Q1:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q2:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q3:

What is the tangent line approximation of near ?

• A
• B
• C
• D
• E

Q4:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q5:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q6:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q7:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q8:

Find the linear approximation of the function at .

Q9:

Find the linear approximation of the function at .

• A
• B0
• C
• D
• E

Q10:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q11:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q12:

Find the linear approximation of the function at .

• A
• B
• C
• D
• E

Q13:

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

• A15.984
• B16.016
• C15.992
• D15.968
• E16.032

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 .

• A
• B
• C
• D
• E

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 ?

• A
• B
• C
• D
• E

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

• A
• B
• C
• D
• E

Determine the value of for which

• A3.8
• B1
• C
• D
• E0

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

• A
• B
• C
• D
• E

Q18:

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

• A
• B2
• C
• D
• E

Q19:

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

• A
• B
• C
• D0
• E

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 .

• A
• B
• C
• D
• E

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 ?

• A
• B
• C
• D
• E

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 ?

• A
• B
• C
• D
• E

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)