Lesson: Linear Approximation

In this lesson, we will learn how to use linear approximation to find the equation of a line that is the closest estimate of a function for a given value of x.

Sample Question Videos

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Worksheet: Linear Approximation • 25 Questions • 3 Videos

Q1:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + 3   at π‘₯ = βˆ’ 2 .

Q2:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = √ π‘₯ at π‘₯ = 4 .

Q3:

What is the tangent line approximation 𝐿 ( π‘₯ ) of √ 1 βˆ’ π‘₯ near π‘₯ = 0 ?

Q4:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = √ π‘₯ 3 at π‘₯ = βˆ’ 8 .

Q5:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ π‘₯ + 1 at π‘₯ = 1 .

Q6:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ s i n at π‘₯ = πœ‹ 6 .

Q7:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ π‘₯ s i n at π‘₯ = 2 πœ‹ .

Q8:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ s i n 2 at π‘₯ = 0 .

Q9:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ t a n at π‘₯ = πœ‹ .

Q10:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = 2 π‘₯ at π‘₯ = 0 .

Q11:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = ( 1 + π‘₯ ) π‘˜ at π‘₯ = 0 .

Q12:

Find the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ s i n βˆ’ 1 at π‘₯ = 0 .

Q13:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ 4 at a suitable value of π‘₯ , estimate the value of ( 1 . 9 9 9 ) 4 .

Q14:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = √ π‘₯ at a suitable value of π‘₯ , estimate the value of √ 1 0 0 . 5 .

Q15:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = 𝑒 π‘₯ at a suitable value of π‘₯ , estimate the value of 𝑒 0 . 1 .

Q16:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = √ π‘₯ 3 at a suitable value of π‘₯ , estimate the value of 3 √ 1 0 0 1 .

Q17:

We will explore why we can call the tangent line approximation the β€œbest” local linearization.

What is the tangent line approximation at π‘₯ = πœ‹ for the function 𝑓 ( π‘₯ ) = 3 . 8 ( π‘₯ ) s i n ?

Suppose 𝐿 ( π‘₯ ) = π‘˜ ( π‘₯ βˆ’ πœ‹ ) is used as a local linearization at π‘₯ = πœ‹ of 𝑓 ( π‘₯ ) = 3 . 8 ( π‘₯ ) s i n . Write the expression for the error 𝐸 ( π‘₯ ) .

Determine the value of π‘˜ for which

Suppose that 𝑓 is a function that is differentiable at π‘₯ = π‘Ž . Using the local linearization at π‘₯ = π‘Ž given by 𝑓 ( π‘₯ ) β‰ˆ 𝑓 ( π‘Ž ) + π‘˜ ( π‘₯ βˆ’ π‘Ž ) , determine

Q18:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ c o s at a suitable value of π‘₯ , estimate the value of c o s 2 9 ∘ .

Q19:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = π‘₯ s i n at a suitable value of π‘₯ , estimate the value of s i n ( 3 . 1 4 ) .

Q20:

Use a local linearization near πœ‹ 2 to estimate 𝑓 ο€» √ 2  to 3 decimal places, where 𝑓 ( π‘₯ ) = 2 π‘₯ s i n .

Q21:

By finding the linear approximation of the function 𝑓 ( π‘₯ ) = 1 π‘₯ at a suitable value of π‘₯ , estimate the value of 1 4 . 0 0 2 .

Q22:

Find the local linearization of 1 √ 1 + π‘₯ near π‘₯ = 0 .

Q23:

The first condition for a linear function 𝐿 ( π‘₯ ) to be an approximation to 𝑓 ( π‘₯ ) near π‘₯ = π‘Ž is that 𝐿 ( π‘Ž ) = 𝑓 ( π‘Ž ) . So 𝐿 ( π‘₯ ) = 𝑓 ( π‘Ž ) + π‘š ( π‘₯ βˆ’ π‘Ž ) for some constant π‘š . The error in using 𝐿 instead of 𝑓 at a point π‘₯ is the function 𝐸 ( π‘₯ ) = 𝑓 ( π‘₯ ) βˆ’ 𝐿 ( π‘₯ ) .

The tangent line approximation to 𝑓 near π‘₯ = π‘Ž is the linear approximation 𝐿 ( π‘₯ ) = 𝑓 ( π‘Ž ) + 𝑓 β€² ( π‘Ž ) ( π‘₯ βˆ’ π‘Ž ) . What is the tangent line approximation to 𝑓 ( π‘₯ ) = 𝑒 π‘˜ π‘₯ near π‘₯ = 0 ?

What is the error in using the tangent line approximation to 𝑓 ( π‘₯ ) = 𝑒 4 . 5 π‘₯ near π‘₯ = 0 at the point 0.1? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to 𝑓 ( π‘₯ ) = 𝑒 4 . 5 π‘₯ near π‘₯ = 0 at the point 0.01? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to 𝑓 ( π‘₯ ) = 𝑒 4 . 5 π‘₯ near π‘₯ = 0 at the point 0.001? Give your answer to 5 decimal places.

Q24:

The curve 𝐢 of equation 𝑦 = 𝑒 π‘₯ is concave up everywhere. The line 𝑦 = 0 . 1 2 π‘₯ + 1 . 0 3 is above 𝐢 when π‘₯ = 0 but then must quickly cross 𝐢 at some 𝛿 > 0 near 0.

What equation can we use to find 𝛿 ?

Find an estimate of 𝛿 using the tangent line approximation of 𝑓 ( π‘₯ ) = 𝑒 π‘₯ at π‘₯ = 0 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 𝑓 ( π‘₯ ) = 𝑒 βˆ’ 2 ( π‘₯ + 1 ) π‘₯ = 1    n e a r ?

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