MCV4U: Calculus & Vectors
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Course Outline
Course Outline -
Unit 1: Rates of ChangeLesson 1: Reviewing prerequisite skills1 Quiz
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Lesson 2: Determining rates of change1 Quiz
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Lesson 3: Determining limits1 Quiz
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Lesson 4: Using first principles to determine the equation of a tangent1 Quiz
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Unit 2 : DerivativesLesson 5: Finding derivatives (part A)1 Quiz
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Lesson 6: Finding Derivatives (Part B)1 Quiz
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Lesson 7: Solving Related Rates Problems1 Quiz
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Lesson 8: Investigating velocity, acceleration and second derivatives1 Quiz
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Unit 3 : Curve SketchingLesson 9: Exploring the first derivative1 Quiz
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Lesson 10: Exploring the second derivative1 Quiz
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Lesson 11: Sketching curves: part A1 Quiz
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Lesson 12: Sketching curves: part B1 Quiz
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Unit 4 : ExtensionsLesson 13: Solving optimization problems1 Quiz
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Lesson 14: Working with sinusoidal functions1 Quiz
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Lesson 15: Working with exponential and logarithmic functions1 Quiz
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Unit 5 : VectorsLesson 16: Using Geometric Vectors1 Quiz
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Lesson 17: Investigating Cartesian vectors1 Quiz
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Lesson 18: Exploring vectors in 3-space1 Quiz
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Lesson 19: Creating equations of vectors1 Quiz
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Lesson 20: Investigating lines and planes1 Quiz
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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,