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.

  • Aπ‘₯βˆ’8
  • B6π‘₯+3
  • C16π‘₯+23
  • D16π‘₯+32
  • E16π‘₯βˆ’9

Q2:

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

  • Aπ‘₯βˆ’2
  • B14π‘₯+1
  • C14π‘₯βˆ’1
  • D14π‘₯+2
  • E12π‘₯

Q3:

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

  • A1βˆ’βˆšπ‘₯2
  • B1+π‘₯2
  • C1βˆ’π‘₯2
  • D1βˆ’π‘₯2
  • E1+π‘₯2

Q4:

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

  • A112π‘₯+23
  • Bβˆ’2π‘₯βˆ’19112
  • C14π‘₯βˆ’4
  • D112π‘₯βˆ’2
  • E112π‘₯βˆ’43

Q5:

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

  • A14π‘₯+14
  • B14π‘₯+12
  • C12π‘₯βˆ’14
  • D14π‘₯βˆ’14
  • E12π‘₯

Q6:

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

  • Aβˆ’βˆš32π‘₯+√3πœ‹12+12
  • Bπ‘₯βˆ’πœ‹6+12
  • C√32π‘₯βˆ’βˆš3πœ‹12+12
  • D√32π‘₯βˆ’βˆš3πœ‹12
  • E√32π‘₯+12

Q7:

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

  • A2πœ‹π‘₯+4πœ‹οŠ¨
  • Bβˆ’2πœ‹π‘₯+4πœ‹οŠ¨
  • Cπ‘₯βˆ’2πœ‹
  • D2πœ‹π‘₯βˆ’4πœ‹οŠ¨
  • E2πœ‹π‘₯

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‹.

  • Aπ‘₯
  • B0
  • Cπ‘₯+πœ‹
  • Dπ‘₯βˆ’πœ‹
  • Eπœ‹βˆ’π‘₯

Q10:

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

  • Aπ‘₯+2ln
  • Bπ‘₯2+1ln
  • Cπ‘₯2ln
  • Dπ‘₯2+1ln
  • Eπ‘₯+1

Q11:

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

  • A1βˆ’π‘˜π‘₯
  • Bπ‘₯+1
  • Cπ‘˜π‘₯+1
  • Dπ‘₯+π‘˜
  • Eπ‘˜π‘₯

Q12:

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

  • Aπ‘₯
  • Bπ‘₯+2πœ‹
  • Cβˆ’π‘₯
  • Dπ‘₯+πœ‹
  • Eπ‘₯+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.

  • A30130
  • B1,001100
  • C3,001300
  • D2,999300
  • E29930

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?

  • A𝐿(π‘₯)=βˆ’3.8(π‘₯+πœ‹)
  • B𝐿(π‘₯)=βˆ’3.8(π‘₯βˆ’πœ‹)
  • C𝐿(π‘₯)=βˆ’3.8
  • D𝐿(π‘₯)=π‘₯βˆ’πœ‹
  • E𝐿(π‘₯)=3.8(π‘₯βˆ’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‘₯)sin. Write the expression for the error 𝐸(π‘₯).

  • A𝐸(π‘₯)=3.8(π‘₯)βˆ’π‘˜(π‘₯βˆ’πœ‹)sin
  • B𝐸(π‘₯)=π‘˜(π‘₯βˆ’πœ‹)βˆ’3.8(π‘₯)sin
  • C𝐸(π‘₯)=3.8(π‘₯βˆ’πœ‹)βˆ’π‘˜(π‘₯βˆ’πœ‹)sin
  • D𝐸(π‘₯)=3.8(π‘₯βˆ’πœ‹)+π‘˜(π‘₯βˆ’πœ‹)sin
  • E𝐸(π‘₯)=π‘˜(π‘₯βˆ’πœ‹)

Determine the value of π‘˜ for which limο—β†’οŽ„πΈ(π‘₯)π‘₯βˆ’πœ‹=0.

  • A3.8
  • B1
  • Cβˆ’3.8
  • Dβˆ’1
  • E0

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.

  • A𝑓′(π‘Ž)
  • B𝑓(π‘Ž)βˆ’π‘˜
  • Cπ‘˜
  • D𝑓(π‘Ž)+π‘˜
  • E𝑓′(π‘Ž)βˆ’π‘˜

Q18:

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

  • A12+√3πœ‹360
  • B2
  • C√32βˆ’πœ‹360
  • D√32+πœ‹360
  • E12βˆ’βˆš3πœ‹360

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).

  • A3.14βˆ’πœ‹
  • Bβˆ’1
  • Cπœ‹+3.14
  • D0
  • Eπœ‹βˆ’3.14

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.

  • A1βˆ’π‘₯
  • B1+π‘₯2
  • C1+π‘₯2
  • D1βˆ’π‘₯2
  • E1+π‘₯

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?

  • A𝐿(π‘₯)=π‘˜π‘₯
  • B𝐿(π‘₯)=1+π‘₯
  • C𝐿(π‘₯)=π‘˜+π‘₯
  • D𝐿(π‘₯)=1
  • E𝐿(π‘₯)=1+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?

  • A𝑒=𝛿
  • B𝑒=1.03
  • C𝑒=0.12𝛿+1.03
  • D𝑒=0.12𝛿
  • E𝑒=0.12π›Ώβˆ’1.03

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)οŠ¨ο—οŠ© near π‘₯=1?

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

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