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(π‘₯) is a function for four known values, where 𝑓(2)=3, 𝑓(6)=3.75, 𝑓(7)=4, and 𝑓(11)=4.25, estimate 𝑓(7).

Q2:

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

𝑑1246
Temperature22242829

Q3:

Use the table to estimate 𝑓(6).

π‘₯2468
𝑓(π‘₯)3.753.94.254.8

Q4:

For the given graph, estimate 𝑓′(βˆ’1.5).

  • A0
  • B1.5
  • Cβˆ’1
  • Dβˆ’1.5
  • E1

Q5:

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

π‘₯12345
𝑓(π‘₯)2.53.253.944.6

Q6:

Given that 𝑦=𝑓(π‘₯) is a function for three known values, where 𝑓(1)=1.5, 𝑓(2)=2.75, and 𝑓(3)=3.25, estimate 𝑓′(2).

Q7:

Use the table to estimate 𝑓′(6).

π‘₯25689
𝑓(π‘₯)12367

Q8:

For the given graph, estimate 𝑓′(3).

  • A1.5
  • B2
  • Cβˆ’2
  • Dβˆ’1.5
  • E3

Q9:

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

Time (Hours)012345
Distance (Kilometers)09098103106112
  • A
    Time (h)012345
    𝑑′(𝑑) (km/h)9090853112
  • B
    Time (h)012345
    𝑑′(𝑑) (km/h)09098103106112
  • C
    Time (h)012345
    𝑑′(𝑑) (km/h)9098138912
  • D
    Time (h)012345
    𝑑′(𝑑) (km/h)90496.544.56
  • E
    Time (h)012345
    𝑑′(𝑑) (km/h)90853112112

Q10:

For the given graph, estimate 𝑓′(2).

  • A4
  • Bβˆ’8
  • Cβˆ’0.5
  • D0.5
  • E8

Q11:

Use the graph of 𝑦=𝑓(π‘₯) to estimate the value of 𝑓′(1.4).

Q12:

Olivia is estimating derivatives for the curve 𝑦=𝑓(π‘₯). She has drawn the following graph.

Use Olivia’s graph to estimate 𝑓′(βˆ’1).

  • A3
  • Bβˆ’1
  • Cβˆ’3
  • Dβˆ’6
  • E6

Q13:

By finding the average of the right and left approximations for the derivative, use the function table to estimate the value of 𝑓′(1).

π‘₯βˆ’1136911151925293234
𝑓(π‘₯)βˆ’84βˆ’76βˆ’64βˆ’53βˆ’20βˆ’12βˆ’11βˆ’9βˆ’8βˆ’2βˆ’17
  • A4
  • Bβˆ’5
  • Cβˆ’76
  • D6
  • E5

Q14:

Use the function table provided to estimate the value of 𝑓′(4).

π‘₯358101214192224
𝑓(π‘₯)46711114221012
  • A821
  • Bβˆ’1
  • C5
  • D1
  • E2

Q15:

Chloe and Jacob want to approximate some derivatives for the function 𝑦=𝑓(π‘₯).

Chloe draws the graph of 𝑦=𝑓(π‘₯) along with tangent lines at the points where π‘₯=βˆ’1, π‘₯=0, π‘₯=1, and π‘₯=2.

Jacob creates the function table for 𝑦=𝑓(π‘₯).

π‘₯βˆ’1012
𝑓(π‘₯)βˆ’1βˆ’2βˆ’32

Chloe and Jacob each approximate the derivative 𝑓′(βˆ’1). Chloe uses her graph, and Jacob uses his table of values. At the end points, Jacob uses right or left approximations; otherwise, he uses symmetric approximations.

State the value of each of their approximations.

  • A1, βˆ’12
  • B1, βˆ’1
  • C1, 1
  • D1, 12
  • Eβˆ’1, βˆ’1

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