# Worksheet: Estimating Derivatives

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

Q1:

Given that is a function for four known values, where , , , and , estimate .

Q2:

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

 𝑡 Temperature 1 2 4 6 22 24 28 29

Q3:

Use the table to estimate .

 𝑥 2 4 6 8 𝑓(𝑥) 3.75 3.9 4.25 4.8

Q4:

For the given graph, estimate . • A0
• B1.5
• C
• D
• E1

Q5:

Use the table to estimate . Take the average of the right and left derivatives.

 𝑥 𝑓(𝑥) 1 2 3 4 5 2.5 3.25 3.9 4 4.6

Q6:

Given that is a function for three known values, where , , and , estimate .

Q7:

Use the table to estimate .

 𝑥 𝑓(𝑥) 2 5 6 8 9 1 2 3 6 7

Q8:

For the given graph, estimate . • A1.5
• B2
• C
• D
• E3

Q9:

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

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

Q10:

For the given graph, estimate . • A4
• B
• C
• D0.5
• E8

Q11:

Use the graph of to estimate the value of . Q12:

Olivia is estimating derivatives for the curve . She has drawn the following graph. Use Olivia’s graph to estimate .

• A3
• B
• C
• D
• E6

Q13:

By finding the average of the right and left approximations for the derivative, use the function table to estimate the value of .

 𝑥 𝑓(𝑥) −1 1 3 6 9 11 15 19 25 29 32 34 −84 −76 −64 −53 −20 −12 −11 −9 −8 −2 −1 7
• A4
• B
• C
• D6
• E5

Q14:

Use the function table provided to estimate the value of ).

 𝑥 𝑓(𝑥) 3 5 8 10 12 14 19 22 24 4 6 7 1 11 14 22 10 12
• A
• B
• C5
• D1
• E2

Q15:

Chloe and Jacob want to approximate some derivatives for the function .

Chloe draws the graph of along with tangent lines at the points where , , , and . Jacob creates the function table for .

 𝑥 𝑓(𝑥) −1 0 1 2 −1 −2 −3 2

Chloe and Jacob each approximate the derivative . 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,
• B1,
• C1, 1
• D1,
• E,