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 𝑓 ( π‘₯ ) = π‘₯ βˆ’ π‘₯ + 3   at π‘₯ = βˆ’ 2 .

  • A 6 π‘₯ + 3
  • B 1 6 π‘₯ + 3 2
  • C 1 6 π‘₯ βˆ’ 9
  • D 1 6 π‘₯ + 2 3
  • E π‘₯ βˆ’ 8

Q2:

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

  • A 1 2 π‘₯
  • B 1 4 π‘₯ βˆ’ 1
  • C 1 4 π‘₯ + 2
  • D 1 4 π‘₯ + 1
  • E π‘₯ βˆ’ 2

Q3:

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

  • A 1 βˆ’ π‘₯ 2
  • B 1 + π‘₯ 2
  • C 1 + π‘₯ 2
  • D 1 βˆ’ π‘₯ 2
  • E 1 βˆ’ √ π‘₯ 2

Q4:

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

  • A 1 4 π‘₯ βˆ’ 4
  • B 1 1 2 π‘₯ + 2 3
  • C 1 1 2 π‘₯ βˆ’ 2
  • D 1 1 2 π‘₯ βˆ’ 4 3
  • E βˆ’ 2 π‘₯ βˆ’ 1 9 1 1 2

Q5:

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

  • A 1 2 π‘₯
  • B 1 4 π‘₯ βˆ’ 1 4
  • C 1 4 π‘₯ + 1 2
  • D 1 4 π‘₯ + 1 4
  • E 1 2 π‘₯ βˆ’ 1 4

Q6:

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

  • A βˆ’ √ 3 2 π‘₯ + √ 3 πœ‹ 1 2 + 1 2
  • B √ 3 2 π‘₯ βˆ’ √ 3 πœ‹ 1 2
  • C √ 3 2 π‘₯ + 1 2
  • D √ 3 2 π‘₯ βˆ’ √ 3 πœ‹ 1 2 + 1 2
  • E π‘₯ βˆ’ πœ‹ 6 + 1 2

Q7:

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

  • A βˆ’ 2 πœ‹ π‘₯ + 4 πœ‹ 2
  • B 2 πœ‹ π‘₯
  • C 2 πœ‹ π‘₯ + 4 πœ‹ 2
  • D 2 πœ‹ π‘₯ βˆ’ 4 πœ‹ 2
  • E π‘₯ βˆ’ 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‹ .

  • A πœ‹ βˆ’ π‘₯
  • B π‘₯
  • C π‘₯ + πœ‹
  • D π‘₯ βˆ’ πœ‹
  • E0

Q10:

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

  • A π‘₯ + 1
  • B π‘₯ 2 l n
  • C π‘₯ 2 + 1 l n
  • D π‘₯ 2 + 1 l n
  • E π‘₯ + 2 l n

Q11:

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

  • A π‘₯ + π‘˜
  • B π‘˜ π‘₯
  • C 1 βˆ’ π‘˜ π‘₯
  • D π‘˜ π‘₯ + 1
  • E π‘₯ + 1

Q12:

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

  • A π‘₯ + 2 πœ‹
  • B π‘₯ + πœ‹
  • C βˆ’ π‘₯
  • D π‘₯
  • E π‘₯ + πœ‹ 2

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 .

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

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 .

  • A 3 0 1 3 0
  • B 1 0 0 1 1 0 0
  • C 2 9 9 9 3 0 0
  • D 3 0 0 1 3 0 0
  • E 2 9 9 3 0

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 ?

  • A 𝐿 ( π‘₯ ) = π‘₯ βˆ’ πœ‹
  • B 𝐿 ( π‘₯ ) = βˆ’ 3 . 8
  • C 𝐿 ( π‘₯ ) = 3 . 8 ( π‘₯ βˆ’ πœ‹ )
  • D 𝐿 ( π‘₯ ) = βˆ’ 3 . 8 ( π‘₯ βˆ’ πœ‹ )
  • E 𝐿 ( π‘₯ ) = βˆ’ 3 . 8 ( π‘₯ + πœ‹ )

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‘₯ ) .

  • A 𝐸 ( π‘₯ ) = 3 . 8 ( π‘₯ ) βˆ’ π‘˜ ( π‘₯ βˆ’ πœ‹ ) s i n
  • B 𝐸 ( π‘₯ ) = 3 . 8 ( π‘₯ βˆ’ πœ‹ ) βˆ’ π‘˜ ( π‘₯ βˆ’ πœ‹ ) s i n
  • C 𝐸 ( π‘₯ ) = π‘˜ ( π‘₯ βˆ’ πœ‹ )
  • D 𝐸 ( π‘₯ ) = π‘˜ ( π‘₯ βˆ’ πœ‹ ) βˆ’ 3 . 8 ( π‘₯ ) s i n
  • E 𝐸 ( π‘₯ ) = 3 . 8 ( π‘₯ βˆ’ πœ‹ ) + π‘˜ ( π‘₯ βˆ’ πœ‹ ) s i n

Determine the value of π‘˜ for which

  • A βˆ’ 1
  • B3.8
  • C βˆ’ 3 . 8
  • D0
  • E1

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

  • A 𝑓 β€² ( π‘Ž ) βˆ’ π‘˜
  • B 𝑓 ( π‘Ž ) βˆ’ π‘˜
  • C π‘˜
  • D 𝑓 β€² ( π‘Ž )
  • E 𝑓 ( π‘Ž ) + π‘˜

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 ∘ .

  • A 1 2 βˆ’ √ 3 πœ‹ 3 6 0
  • B2
  • C √ 3 2 βˆ’ πœ‹ 3 6 0
  • D √ 3 2 + πœ‹ 3 6 0
  • E 1 2 + √ 3 πœ‹ 3 6 0

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 ) .

  • A0
  • B βˆ’ 1
  • C πœ‹ + 3 . 1 4
  • D πœ‹ βˆ’ 3 . 1 4
  • E 3 . 1 4 βˆ’ πœ‹

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 .

  • A 1 + π‘₯ 2
  • B 1 βˆ’ π‘₯
  • C 1 + π‘₯
  • D 1 βˆ’ π‘₯ 2
  • E 1 + π‘₯ 2

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 ?

  • A 𝐿 ( π‘₯ ) = 1
  • B 𝐿 ( π‘₯ ) = 1 + π‘₯
  • C 𝐿 ( π‘₯ ) = π‘˜ + π‘₯
  • D 𝐿 ( π‘₯ ) = 1 + π‘˜ π‘₯
  • E 𝐿 ( π‘₯ ) = π‘˜ π‘₯

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 ?

  • A 𝑒 = 0 . 1 2 𝛿 𝛿
  • B 𝑒 = 𝛿 𝛿
  • C 𝑒 = 1 . 0 3 𝛿
  • D 𝑒 = 0 . 1 2 𝛿 + 1 . 0 3 𝛿
  • E 𝑒 = 0 . 1 2 𝛿 βˆ’ 1 . 0 3 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 ?

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

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