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MCV4U: Calculus & Vectors

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  1. Course Outline

    Course Outline

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  3. Unit 1: Rates of Change
    Lesson 1: Reviewing prerequisite skills
    1 Quiz

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  5. Lesson 2: Determining rates of change
    1 Quiz

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  7. Lesson 3: Determining limits
    1 Quiz

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  9. Lesson 4: Using first principles to determine the equation of a tangent
    1 Quiz

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  11. Unit 2 : Derivatives
    Lesson 5: Finding derivatives (part A)
    1 Quiz

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  13. Lesson 6: Finding Derivatives (Part B)
    1 Quiz

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  15. Lesson 7: Solving Related Rates Problems
    1 Quiz

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  17. Lesson 8: Investigating velocity, acceleration and second derivatives
    1 Quiz

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  19. Unit 3 : Curve Sketching
    Lesson 9: Exploring the first derivative
    1 Quiz

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  21. Lesson 10: Exploring the second derivative
    1 Quiz

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  23. Lesson 11: Sketching curves: part A
    1 Quiz

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  25. Lesson 12: Sketching curves: part B
    1 Quiz

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  27. Unit 4 : Extensions
    Lesson 13: Solving optimization problems
    1 Quiz

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  29. Lesson 14: Working with sinusoidal functions
    1 Quiz

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  31. Lesson 15: Working with exponential and logarithmic functions
    1 Quiz

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  33. Unit 5 : Vectors
    Lesson 16: Using Geometric Vectors
    1 Quiz

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  35. Lesson 17: Investigating Cartesian vectors
    1 Quiz

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  37. Lesson 18: Exploring vectors in 3-space
    1 Quiz

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  39. Lesson 19: Creating equations of vectors
    1 Quiz

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  41. 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 ‚Äď a or x = a + 

The coordinate Q can be expressed as Q(a + , f(a + )) and  = f(a + ) ‚Äď 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 ‚Äúf prime of a‚ÄĚ 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 ‚Äď4.¬† 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 ‚Äďintercept is ‚Äď7.¬†