Worksheet: Estimating Derivatives

In this worksheet, we will practice using numerical and graphical methods to estimate the derivative of a function.

Q1:

Given that 𝑦 = 𝑓 ( 𝑥 ) is a function for four known values, where 𝑓 ( 2 ) = 3 , 𝑓 ( 6 ) = 3 . 7 5 , 𝑓 ( 7 ) = 4 , and 𝑓 ( 1 1 ) = 4 . 2 5 , estimate 𝑓 ( 7 ) .

  • A0.25
  • B0.3125
  • C0.15625
  • D0.1875
  • E0.0625

Q2:

On a particular day in a city, the temperature is measured, in degrees Celsius, every 𝑡 hours. Using the given table, estimate the rate of change in the temperature at 𝑡 = 4 .

𝑡 1 2 4 6
Temperature 22 24 28 29

Q3:

Use the table to estimate 𝑓 ( 6 ) .

𝑥 2 4 6 8
𝑓 ( 𝑥 ) 3.75 3.9 4.25 4.8
  • A0.45
  • B0.1
  • C0.275
  • D0.225
  • E0.175

Q4:

For the given graph, estimate 𝑓 ( 1 . 5 ) .

  • A1.5
  • B 1
  • C 1 . 5
  • D0
  • E1

Q5:

Use the table to estimate 𝑓 ( 4 ) . Take the average of the right and left derivatives.

𝑥 1 2 3 4 5
𝑓 ( 𝑥 ) 2.5 3.25 3.9 4 4.6

Q6:

Given that 𝑦 = 𝑓 ( 𝑥 ) is a function for three known values, where 𝑓 ( 1 ) = 1 . 5 , 𝑓 ( 2 ) = 2 . 7 5 , and 𝑓 ( 3 ) = 3 . 2 5 , estimate 𝑓 ( 2 ) .

Q7:

Use the table to estimate 𝑓 ( 6 ) .

𝑥 2 5 6 8 9
𝑓 ( 𝑥 ) 1 2 3 6 7

Q8:

For the given graph, estimate 𝑓 ( 3 ) .

  • A 1 . 5
  • B2
  • C3
  • D 2
  • E1.5

Q9:

The table shows the distance 𝑑 traveled by a car in kilometers after time 𝑡 in hours. Use the information shown to construct a table of values of 𝑑 ( 𝑡 ) .

Time (Hours) 0 1 2 3 4 5
Distance (Kilometers) 0 90 98 103 106 112
  • A
    Time (h) 0 1 2 3 4 5
    𝑑 ( 𝑡 ) (km/h) 90 8 5 3 112 112
  • B
    Time (h) 0 1 2 3 4 5
    𝑑 ( 𝑡 ) (km/h) 0 90 98 103 106 112
  • C
    Time (h) 0 1 2 3 4 5
    𝑑 ( 𝑡 ) (km/h) 90 98 13 8 9 12
  • D
    Time (h) 0 1 2 3 4 5
    𝑑 ( 𝑡 ) (km/h) 90 90 8 5 3 112
  • E
    Time (h) 0 1 2 3 4 5
    𝑑 ( 𝑡 ) (km/h) 90 49 6.5 4 4.5 6

Q10:

For the given graph, estimate 𝑓 ( 2 ) .

  • A 8
  • B 0 . 5
  • C4
  • D8
  • E0.5

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