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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 1: Rates of Change

Unit 1: Rates of Change 

  • ¬†change over an interval and instantaneous rate of change at a point, using the slopes of secants and¬†tangents and the concept of the limit;¬†
  • ¬†between the numeric, graphical, and algebraic representations of a function and its derivative;¬†
  • ¬†graphically and algebraically the rules for determining derivatives; apply these rules to¬†determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions,¬†and simple combinations of functions; and solve related problems.¬†

Lesson 1: Rates of Change 

Refresh Your memory ‚Äď Review 

Following are some topics you will need to know to start this course.  If you are NOT comfortable with these topics, you need to revisit them or take a pre-requisite foundation math course prior to continuing.

  1. Factoring Trinomials

Difference of squares

Sum of Cubes

Difference of Cubes

Factor Theorem: A polynomial P(x) has a factor x ‚Äď a if and only if¬†P(a) = 0.

EXAMPLE: 

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Rationalizing a Denominator

  • the process of changing¬† a denominator form a radical (square root) to a rational number (integer)¬†
  • ¬†reason that we rationalize denominators is that dividing by an integer is preferable to dividing by a radical number.¬†

Note: when the denominator of a radical fraction is a two term expression, you can rationalize the denominator of by multiplying the conjugate.

EXAMPLE:

2. Linear Functions

  • a linear function is a function of the form¬†or¬†, where¬†m¬†= slope,¬†b¬†= y-int¬†of the line¬†
  • the slope of a line that passes through the points¬†P1(x1, y1)¬†and¬†P2(x2, y2)¬†is defined by¬†¬†
  • since¬†the slope is the ratio of the change in¬†y¬†to the change in¬†x, it can be interpreted as the ‚Äúrate of change¬†of¬†y¬†with respect to¬†x‚ÄĚ.¬†
  • Sequences and Series¬†(covered in Grade 11 U and M Mathematics)¬†
  • Slope of¬†the¬†Tangent, Slope of the Secant, Slope of the Tangent at an Arbitrary Point¬†

A tangent is the straight line that most resembles the graph near a point.  Its shape tells how steep the graph is at the pint of tangency.  In the figure below, four tangents have been drawn. 

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To determine the equation of a tangent to a curve at a given point, we fist need to know the slope of the tangent.  When only one point is known, proceed as follows; 

Consider a curvy y=f(x) and a point P that lies on the curve.  Now condister another point Q on the curve.  The ling joining P and Q is called a secant.  Think of Q as a moving point that slides along the curve towards P, so that the slope of the secant PQ becomes a progressively better estimate of the slope of the tangent at P

Therefore, the definition for the slope of the tangent might be; 

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The slope of the tangent to a curve at a point P is the limiting slope of the secant PQ as the point Q slides along the curve towards P.  In other words, the slope of the tangent is said to be the limit of the slope of the secant as Q approaches P along the curve. 

The slope of the tangent to the graph y=f(x) at point P(a, f(a)), is 

Rates of Change

Recall the slope of a line.  Since the slope is the ratio of the change in y to the change in x, it can be interpreted as the rate of change of y with respect to x

Below are real-world examples of rates of change.

EXAMPLE 1 

The boiling point is the temperature at which a liquid changes to gas.  When water is heated it eventually reaches a temperature Рthe boiling point Рat which the vapour pressure is large enough that bubbles are formed inside the body of the water. The boiling point depends on the pressure and the boiling points of water at pressures ranging from 0.5 psia to 1000 psia can be found in the diagram and table below. 

Describe the rate of change that is occurring both on the pressure and the boiling points of water ranging from 0.5 psia to 1000 psia.

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SOLUTION:

The rate of change on the pressure is linear since it is increasing at a constant rate.  The rate of change on the boiling points is non-linear in both degrees F and degrees C because the temperature is rising at a non-constant rate. 

EXAMPLE 2:

When a bathtub is draining, would you expect half of the water to drain in half of the time?  Explain.

Draining Water from a Bathtub

Time(s) Volume(L) 
900 
10 715 
20 551 
30 403 
40 283 
50 179 
60 89 
70 42 
80 12 
90 

SOLUTION:

Water pressure would make the water drain quicker at the beginning than near the end of the draining process.  Therefore, it would drain half of the water in less than half the time.