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MCV4U: Calculus & Vectors

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  1. Course Outline
    Course Outline
  2. Unit 1: Rates of Change
    Lesson 1: Reviewing prerequisite skills
    1 Quiz
  3. Lesson 2: Determining rates of change
    1 Quiz
  4. Lesson 3: Determining limits
    1 Quiz
  5. Lesson 4: Using first principles to determine the equation of a tangent
    1 Quiz
  6. Unit 2 : Derivatives
    Lesson 5: Finding derivatives (part A)
    1 Quiz
  7. Lesson 6: Finding derivatives (part B)
    1 Quiz
  8. Lesson 7: Solving related rates problems
    1 Quiz
  9. Lesson 8: Investigating velocity, acceleration and second derivatives
    1 Quiz
  10. Unit 3 : Curve Sketching
    Lesson 9: Exploring the first derivative
    1 Quiz
  11. Lesson 10: Exploring the second derivative
    1 Quiz
  12. Lesson 11: Sketching curves: part A
    1 Quiz
  13. Lesson 12: Sketching curves: part B
    1 Quiz
  14. Unit 4 : Extensions
    Lesson 13: Solving optimization problems
    1 Quiz
  15. Lesson 14: Working with sinusoidal functions
    1 Quiz
  16. Lesson 15: Working with exponential and logarithmic functions
    1 Quiz
  17. Unit 5 : Vectors
    Lesson 16: Using geometric vectors
    1 Quiz
  18. Lesson 17: Investigating Cartesian vectors
    1 Quiz
  19. Lesson 18: Exploring vectors in 3-space
    1 Quiz
  20. Lesson 19: Creating equations of vectors
    1 Quiz
  21. Lesson 20: Investigating lines and planes
    1 Quiz
Lesson 22 of 21
In Progress

Lesson 3: Average & Instantaneous Rates of Change

Average Rate of Change 

The table shows the results of a student recording temperature every 3 s.   

Time(s) Temperature (F) 
66.756 
89.330 
91.544 
92.678 
12 93.542 
15 94.010 
18 94.496 
21 94.550 

How can you estimate the rate of change of temperature at exactly 15 s?   

A closer look at the temperatures in the table shows that for every 3 s the temperature is recorded from 0 to 21 s to the nearest thousandths.  The temperature amounts are increasing but are not a constant rate between time intervals. 

In other words, the rate of change of temperature is different for different intervals. 

The table below shows the first difference and the average rate of change for each 3 s interval.   

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The average rate of change appears to decrease as time increases.  The average rate of change is  from 12 to 15 s and the average rate of change is  from 15 to 18s.  So it is difficult to estimate the rate of change at exactly 15 s, however, a sequence of average rates of change can be used to estimate the rate of change at an exact time. 

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Solution:

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b. The time interval is 1< t > 2. 

The average rate of change of height with respect to time is 3.9 m/s during the 2nd second of the modelā€™s flight.