This comprehensive document delves into the fundamental concepts and principles of calculus, providing a detailed exploration of the subject's core topics. With clear explanations and concise examples, it covers:
- Limits and continuity
- Derivatives and differentiation rules
- Applications of...
Objectives:
✠ Introduces a technique called differentiation for calculating
the gradient of a curve at any point
✠ Introduces some rules for finding gradient functions
✠ Explains what is meant by the terms ”first derivative” and
”second derivative”
✠ explains the terms ”maximum” and ”minimum” when
applied to functions
✠ Applies the technique of differentiation to locating
maximum and minimum values of a function
, Overview of Differential Calculus
⋇ Differential calculus along with Integral calculus are the
two branches of Calculus, the study of continuous change
or a rate of change of a function.
⋇ Calculus was developed by Newton (1642 - 1727) and
Leibnitz (1646 - 1716) to deal with finding the different
properties of derivatives and intergrals of a function.
⋇ Differential calculus deals with the rate of change of one
quantity with respect to another quantity.
, Overview of differential Calculus Cont....
Gradient of a curve
⋇ The gradient of a straight line is a constant and simply
∆
determined by ∆xy which is the ratio between any two
points on the line
⋇ But, on the curve the gradient is changing from one point
to another
⋇ Thus, we can define the gradient at any point on a curve
to be the gradient of the tangent to the curve at that
point.
⋇ Recall that a tangent to a curve is a straight line that
touches the curve at one point.
⋇ To find a gradient at any point on the curve, we can make
use of the method called a limiting process, sometimes
known as differentiation from the first principle.
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