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# ACMMM080

## 1.2 Derivatives Basic Concepts – Tangent and Derivatives

#### Chord, Secant, and Tangent

• A chord is a line segment joining two points on a curve, like the line AB in the previous example.
• A secant is a line that passes through two points, which means that it can be longer than a chord for any given two points A and B.
• Graphically, ‘a tangent though a curve at point P’ is defined as a line that passes through P and its gradient equals the curve’s instantaneous rate of change at P.
• Suppose P has coordinates P(p,\:f(p)), then \text {Gradient of tangent at } P=\lim _{h \rightarrow 0} \frac{f(p+h)-f(p)}{h}

Note: Here, h \rightarrow 0 is used and not h \rightarrow 0^{+}.

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## 1.1 Derivatives Basic Concepts – Rate of Changes

#### Average Rate of Change

• The average change of y with respect to x over the interval [a, b] is the gradient of the line connecting the points A(a,f(a)) and B(b,f(b)).

Note: [a,b] represents a range between a and b, inclusive. In other words, if x \in[a, b] , then \mid a \leq x \leq b . Also, we will always have a<b .

• It is given by:
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