Yesterday I have managed to complete my next chapter of Brief Calculus & Its Applications.

The following topics were covered by Chapter 1:

- The Slope of a Straight Line
- The Slope of a Curve at a Point
- The Derivative and Limits
- Limits and the Derivative
- Differentiability and Continuity
- Some Rules for Differentiation
- More About Derivatives
- The Derivative as a Rate of Change

So far so good but I guess my pace is to high right now. I will spend some time going through the book once again just in case I missed anything.

~450 pages left to finish. I might slow down a little bit but hope to be done with the book by the end of next week.

A little bit of theory. Lets start with **The Slope of a Straight Line**:

We can compute the slope of a line by knowing two points on the line. If (x1,y1) and (x2,y2) are on the line, the slope of the line (m) is:

`m = (y2 − y1) / (x2 −x1)`

**The slope of a curve at a point P** is defined to be the slope of the tangent line to the curve at P and follows the slope formula

` slope of the graph of y = x^2 at the point (x,y) = 2x`

**The slope formula** that gives the **slope of the curve y = f(x) at any point** is called the **derivative** of f(x) and is written f′(x). In other words – the derivative f′(a) **measures the rate of change** of f(x) at x = a.

There are 3 rules we need to remember:

- If f(x) = mx + b, then we have f′(x) = m
- The derivative of a constant function f(x) = b is zero. That is, f′(x) = 0
- Power rule: let r be any number and let f(x) = x^r. Then f′(x) = rx^(r−1)

Examples of **Power rule**

- If f (x) = x^2 , then its derivative is the function 2x. That is, f′(x) = 2x
- If f (x) = x^3 , then the derivative is 3x^2 . That is, f′(x) = 3x^2
- If f (x) = 1, then f′(x)=−1 (x̸0)

And now we can also write **Equation of the Tangent Line**

`y − f(a) = f′(a)(x − a)`

Another important thing to remember is that on the tangent line, **the change in y, for one unit change in x, is equal to the slope f′(a)**

`f(a + 1) − f(a) ≈ f′(a) OR f(a + 1) ≈ f(a) + f′(a) `

And to summarise everything

### Like this:

Like Loading...