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 π‘₯ βˆ’ 8
  • E 1 6 π‘₯ + 2 3

Q2:

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

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

Q3:

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

  • A 1 + π‘₯ 2
  • B 1 βˆ’ 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‘₯ at π‘₯=βˆ’8.

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

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

Q9:

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

  • A π‘₯ βˆ’ πœ‹
  • B πœ‹ βˆ’ π‘₯
  • C0
  • D π‘₯
  • E π‘₯ + πœ‹

Q10:

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

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

Q11:

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

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

Q12:

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

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

Q13:

By finding the linear approximation of the function 𝑓(π‘₯)=π‘₯οŠͺ at a suitable value of π‘₯, estimate the value of (1.999)οŠͺ.

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

Q14:

By finding the linear approximation of the function 𝑓(π‘₯)=√π‘₯ at a suitable value of π‘₯, estimate the value of √100.5.

Q15:

By finding the linear approximation of the function 𝑓(π‘₯)=𝑒 at a suitable value of π‘₯, estimate the value of π‘’οŠ¦οŽ–οŠ§.

Q16:

By finding the linear approximation of the function 𝑓(π‘₯)=√π‘₯ at a suitable value of π‘₯, estimate the value of √1,001.

  • A 3 0 1 3 0
  • B 1 , 0 0 1 1 0 0
  • C 3 , 0 0 1 3 0 0
  • D 2 , 9 9 9 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‘₯)sin?

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

Suppose 𝐿(π‘₯)=π‘˜(π‘₯βˆ’πœ‹) is used as a local linearization at π‘₯=πœ‹ of 𝑓(π‘₯)=3.8(π‘₯)sin. Write the expression for the error 𝐸(π‘₯).

  • A 𝐸 ( π‘₯ ) = π‘˜ ( π‘₯ βˆ’ πœ‹ )
  • B 𝐸 ( π‘₯ ) = 3 . 8 ( π‘₯ ) βˆ’ π‘˜ ( π‘₯ βˆ’ πœ‹ ) s i n
  • C 𝐸 ( π‘₯ ) = π‘˜ ( π‘₯ βˆ’ πœ‹ ) βˆ’ 3 . 8 ( π‘₯ ) s i n
  • D 𝐸 ( π‘₯ ) = 3 . 8 ( π‘₯ βˆ’ πœ‹ ) βˆ’ π‘˜ ( π‘₯ βˆ’ πœ‹ ) 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 limο—β†’οŽ„πΈ(π‘₯)π‘₯βˆ’πœ‹=0.

  • A βˆ’ 1
  • B3.8
  • C0
  • D βˆ’ 3 . 8
  • 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‰ˆπ‘“(π‘Ž)+π‘˜(π‘₯βˆ’π‘Ž), determine limο—β†’οŒΊπΈ(π‘₯)π‘₯βˆ’π‘Ž.

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

Q18:

By finding the linear approximation of the function 𝑓(π‘₯)=π‘₯cos at a suitable value of π‘₯, estimate the value of cos29∘.

  • A2
  • B 1 2 + √ 3 πœ‹ 3 6 0
  • C √ 3 2 + πœ‹ 3 6 0
  • D 1 2 βˆ’ √ 3 πœ‹ 3 6 0
  • E √ 3 2 βˆ’ πœ‹ 3 6 0

Q19:

By finding the linear approximation of the function 𝑓(π‘₯)=π‘₯sin at a suitable value of π‘₯, estimate the value of sin(3.14).

  • A 3 . 1 4 βˆ’ πœ‹
  • B0
  • C πœ‹ βˆ’ 3 . 1 4
  • D βˆ’ 1
  • E πœ‹ + 3 . 1 4

Q20:

Use a local linearization near πœ‹2 to estimate π‘“ο€»βˆš2 to 3 decimal places, where 𝑓(π‘₯)=2π‘₯sin.

Q21:

By finding the linear approximation of the function 𝑓(π‘₯)=1π‘₯ at a suitable value of π‘₯, estimate the value of 14.002.

Q22:

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

  • A 1 βˆ’ π‘₯ 2
  • B 1 βˆ’ π‘₯
  • C 1 + π‘₯ 2
  • D 1 + π‘₯
  • 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’πΏ(π‘₯).

The tangent line approximation to 𝑓 near π‘₯=π‘Ž is the linear approximation 𝐿(π‘₯)=𝑓(π‘Ž)+𝑓′(π‘Ž)(π‘₯βˆ’π‘Ž). What is the tangent line approximation to 𝑓(π‘₯)=𝑒 near π‘₯=0?

  • A 𝐿 ( π‘₯ ) = π‘˜ + π‘₯
  • B 𝐿 ( π‘₯ ) = 1
  • C 𝐿 ( π‘₯ ) = 1 + 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‘₯)=𝑒οŠͺοŽ–οŠ«ο— 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‘₯)=𝑒οŠͺοŽ–οŠ«ο— 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‘₯)=𝑒οŠͺοŽ–οŠ«ο— near π‘₯=0 at the point 0.001? Give your answer to 5 decimal places.

Q24:

The curve 𝐢 of equation 𝑦=𝑒 is concave up everywhere. The line 𝑦=0.12π‘₯+1.03 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 = 𝛿 
  • B 𝑒 = 1 . 0 3 
  • C 𝑒 = 0 . 1 2 𝛿 + 1 . 0 3 
  • D 𝑒 = 0 . 1 2 𝛿 
  • E 𝑒 = 0 . 1 2 𝛿 βˆ’ 1 . 0 3 

Find an estimate of 𝛿 using the tangent line approximation of 𝑓(π‘₯)=𝑒 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‘₯)=π‘’βˆ’2(π‘₯+1)π‘₯=1οŠ¨ο—οŠ©near?

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

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