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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 4: Limits & Instantaneous Rates of Change Derivative

The Derivative at a Point 

Previously it was mentioned that the instantaneous rate of change at point P(a, f(a)) is equal to the slope of the tangent line at that point. 

The slope of the tangent line is the limiting value of the slopes of the secants, represented by PQ, as point Q approaches point P along the curve.   

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 is the interval between the x-coordinates of P and Q.  So, = x – a or x = a + 

The coordinate Q can be expressed as Q(a + , f(a + )) and  = f(a + ) – f(a). 


Therefore the slope of the tangent can be determined as such: 


When we let h = then slope of the tangent becomes

Therefore, the derivative at a point is defined as:

The derivative of a function f at point (a, f(a)) is f1(a) = , if this limit exists.  The notation  is read “f prime of a” and is the slope of the tangent to the function 

Example 1:

Determine the equation of the tangent to the curve defined by  at point (3, -10). 

Solution 

The derivative is used to find the slope of the tangent at ((a, f(a)) = (3, -10). 

Remember that:    so,

At point (3, -10), the slope of the tangent is –4.  Use y = mx + b and substitution to find the equation of the tangent. 


Therefore the equation of the tangent is . 

Example 2:

Sketch the graph of f(x) and the tangent line from example 1. 

Solution:

To get the points for the parabola completing the square is required.  First you must put the quadratic equation in vertex form . 

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The vertex of the parabola is (1, -6) and the y –intercept is –7.