This is the first of three sections on differential calculus. This is significant branch of mathematics with lots of applications. It builds very nicely on other concepts in the course and those you are likely to have covered previously. The whole principle is based around examining 'Rates of change'.
Key Concepts
In this unit you should learn to
Understand the basic principles of differential calculus as rate of change
Differentiate polynomial functions and use this work out gradients
Differentiate functions involving negative indices
Use basic differentiation to solve problems about gradients
Essentials
Slides Gallery
Use these slides to review the material and key points covered in the videos.
1. Introduction - what is calculus about
This video puts the whole idea in to some context and shows how calculus builds carefully on work already done.
2. The Gradient function
Here we introduce the idea of deriving a function which tells about the gradient of the function we started with.
3. Differentiating polynomials
At this point we move on to the general pattern for deriving the gradient function.
4. Negative indices and differentiating
This is a focus on dealing with negative indices. They are a potential pitfall, but if you are careful there should be no problem.
5. Finding gradients
Here we apply the techniques to find the gradient of a given function at any point.
Summary
Review these condensed 'key point' flashcards to help you check and keep ideas fresh in your mind.
Test yourself
Self Checking Quiz
Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.
1
Identify the gradient/rate of change of the following functions
y = 3x +2
y= -5x
y = 7
y = 0.5x - 4
These are all straight line (linear) functions in the form y = mx + c, where m is the gradient. In the case of y = 7, we can think of this as y = 0x + 7 and so the gradient is 0. Graphs like this are horizontal lines so it is logical to conclude that they have gradient = 0.
2
Consider the function for which . What are the values of a and b?
a = , b =
3
Consider the function for which . What are the values of a and b?
a = , b =
4
Consider the function , for which . What are the values of a, b and c
a = , b= , c =
5
Consider the function , for which . What are the values of a, b, c and d
a = , b= , c = , d =
, watch out for signs.
6
can also be written as , what are the values of a, b and c?
, b = , c =
7
Consider the function , for which . What are the values of a, b, c and d? (do it on paper and be careful with the signs)
a = , b= , c = , d =
8
What is the gradient of the function when x = 2?
Gradient =
9
What is the gradient of the function when x = -3?
Gradient =
10
Given find to 3sf
=
159 to 3sf (use the table function on your GDC
Exam Style Questions
The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.
Question 1
Video Solution
Question 2
Video Solution
Question 3
Video Solution
MY PROGRESS
How much of 5.1 & 5.3 Introduction to Calculus have you understood?
Feedback
Which of the following best describes your feedback?