# Derivatives and rates of change

## Slopes of lines and tangent lines

• Recall slopes
• Let $$m$$ be a number, and $$P(a,c)$$ a plane point.
• Then the equation of the line through $$P$$ with slope $$m$$ is the following. $y-c=m(x-a)$
• How you can remember this is that $$y=mx$$ is the line with slope $$m$$ through the origin $$O(0,0)$$, which you then need to translate by $$c$$ upwards, and by $$a$$ to the right.
• Tangent lines
• Let $$f(x)$$ be a function, and let $$C:y=f(x)$$ be the curve which is its graph.
• Let us fix two numbers $$a$$ and $$b$$ such that the closed interval $$[a,b]$$ is contained in the domain of $$f(x)$$, and consider the points $$P(a,f(a))$$ and $$Q(b,f(b))$$ on $$C$$.
• Then the line between $$P$$ and $$Q$$ will have the following slope. $m_{PQ}=\frac{f(b)-f(a)}{b-a}.$
• Definition. Let $$f(x)$$ be a function, and $$a$$ a point in its domain. Then the tangent line to the curve $$C=f(x)$$ at $$P(a,f(a))$$ is the line through $$P$$ with slope $m=\lim_{b\to a}\frac{f(b)-f(a)}{b-a},$ provided that the limit exists.

## Rate of change and derivatives

• Let $$f(x)$$ be a function, and $$a$$ and $$b$$ two numbers such that the closed interval $$[a,b]$$ is contained in the domain of $$f(x)$$.
• We can interpret the difference $$f(b)-f(a)$$ as the average rate of change of $$f(x)$$ on the interval $$[a,b]$$.
• There's another notation emphasising this point of view.
• The change in $$x$$ is $$\Delta x=b-a$$, and the change in $$y=f(x)$$ is $$\Delta y=f(b)-f(a)$$.
• Then the average rate of change can be written as $$\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$$.
• As we make $$b\to a$$, we get the (instantaneous) rate of change at $$x=a$$.
• Since we make $$b\to a$$, it's better to write $$b=a+h$$.
• Note that $$h=\Delta x=b-a$$.
• Definition. Let $$f(x)$$ be a function, and $$a$$ a number such that the domain of $$f(x)$$ contains some open interval around $$a$$. Then the derivative of $$f(x)$$ at $$x=a$$ is $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h},$ provided that the limit exists.
• An alternative notation is $$\frac{\mathrm df}{\mathrm dx}\big|_{x=a}$$.
• Exercises. 2.7: 8ab, 10ab, 14, 20, 22, 60