Back to Course

## MCV4U: Calculus & Vectors

0% Complete
0/0 Steps

Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
1. ### Course Outline

Course Outline

2. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
3. Unit 1: Rates of Change
Lesson 1: Reviewing prerequisite skills
1 Quiz

4. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
5. Lesson 2: Determining rates of change
1 Quiz

6. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
7. Lesson 3: Determining limits
1 Quiz

8. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
9. Lesson 4: Using first principles to determine the equation of a tangent
1 Quiz

10. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
11. Unit 2 : Derivatives
Lesson 5: Finding derivatives (part A)
1 Quiz

12. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
13. Lesson 6: Finding Derivatives (Part B)
1 Quiz

14. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
15. Lesson 7: Solving Related Rates Problems
1 Quiz

16. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
17. Lesson 8: Investigating velocity, acceleration and second derivatives
1 Quiz

18. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
19. Unit 3 : Curve Sketching
Lesson 9: Exploring the first derivative
1 Quiz

20. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
21. Lesson 10: Exploring the second derivative
1 Quiz

22. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
23. Lesson 11: Sketching curves: part A
1 Quiz

24. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
25. Lesson 12: Sketching curves: part B
1 Quiz

26. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
27. Unit 4 : Extensions
Lesson 13: Solving optimization problems
1 Quiz

28. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
29. Lesson 14: Working with sinusoidal functions
1 Quiz

30. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
31. Lesson 15: Working with exponential and logarithmic functions
1 Quiz

32. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
33. Unit 5 : Vectors
Lesson 16: Using Geometric Vectors
1 Quiz

34. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
35. Lesson 17: Investigating Cartesian vectors
1 Quiz

36. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
37. Lesson 18: Exploring vectors in 3-space
1 Quiz

38. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
39. Lesson 19: Creating equations of vectors
1 Quiz

40. Notice: Undefined offset: 0 in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143

Notice: Trying to access array offset on value of type null in /home/forge/cilearning.ca/web/app/themes/buddyboss-theme/learndash/ld30/learndash-sidebar.php on line 143
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.

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

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