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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 2: Understanding Limits

By definition a limit is: A function f  has a limit L as x approaches a, written  provided that the values of f(x) get closer and closer to L as x gets closer and closer to a, on both sides of a, but x ≠ a

Another way to state this definition is to say that a limit is the intended height (where height f is a function in term of time x) of a function

Determining the height of a function

Example 1:

This function changes height all throughout its domain.  That different “x” values have different heights.

What is the height of the above function when x = 2?

Solution:

The function reaches a height of 4 when x = 2.  A limit statement can be made.  which simply means that the function as x approaches 2 is getting closer to the height of 4.

Functions that don’t reach their intended heights

Functions that don’t reach their intended heights

Example 2: Can a function have a limit if it can’t reach its intended height?

This function appears to be a regular function; however, there is a hole in the middle of the line representing this function.

When x =2 the function is undefined.

When we substitute x = 2 into the function the following happens.

When Does a Limit Exist?

A limit exists if you travel along a function from the left side and from the right side towards some specific value of x. In other words, as long as the y-values from the left and right are the same then the limit exists.

Example 3: What is the Limit of the function shown below?

Solution:

For a limit to exist the left hand limit must equal the right hand limit.

So, in this example the limit does not exist.

This is also called a discontinued function.  If the limit did exist it would be called a continuous function.

Discontinued Function:

Continuous Function:

Evaluating Limits (substitution)

Example 4:

Solution:

Substitute 3 in for n.

Therefore,

Note: If you substitute and there is an answer, the limit is the answer.

Example 5:

Solution:

Factor the numerator first then since a substitution yielded an undefined answer, cancel out common factors to the numerator and denominator and substitute.

Therefore,

Example 6: Find

Solution: Conjugates are used to evaluate this question because we must rationalize the numerator (see Lesson 1).

Therefore,