Lecture 1

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Introduction

2 Founders

• Newton, Leibniz

2 Divisions

• Differential Calculus (slope)
• Integral Calculus (area/volume)

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CARTESIAN COORDINATE

• Subdivision of flat plane uses 2 lines at right-angle.
• Subdivide plane into 4 quadrants.

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INCREMENTS

• We use of Greek capital Delta, "$\Delta$" to indicate CHANGE of a variable. For instance, $\Delta{x}$ means "change in $x$".
• Suppose we move $\left(2,\:1\right)$ to $\left(4,\:2\right)$, then

$$\begin{align*} \Delta{x}&=x_{\text{final}}-x_{\text{initial}}=(4)-(2)=2;\\ \Delta{y}&=y_{\text{final}}-y_{\text{initial}}=(2)-(1)=1; \end{align*}$$

In general, if we move from point $P_1\left(x_1,\:y_1\right)$ to point $P_2\left(x_2,\:y_2\right)$, then

$$\begin{align*} \Delta{x}&=x_2-x_1\\ \Delta{y}&=y_2-y_1 \end{align*}$$

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SLOPE OF STRAIGHT LINE

• Slope = steepness
• Definition of slope:

$$\begin{align*} m&=\frac{\Delta{y}}{\Delta{x}}\\ &=\frac{y_2-y_1}{x_2-x_1}\\ &=\frac{\text{rise}}{\text{run}} \end{align*}$$

EXAMPLE: SLOPE OF STRAIGHT LINE

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REMARKS

Remark 1

On a straight line, slope is the same regardless of which points are used to compute at.

Remark 2

We can interpret slope in

• Proportionality factor, i.e. $\Delta{y}=m\Delta{x}$
• The angle of indication that a line makes with the $x$-axis (or any horizontal line).

Implication

• If 2 lines have the same angle of inclination, then the slope are same and the lines are parallel.
• If the line slopes downward, the $\tan{\left(\theta\right)}$ is negative. Therefore the slope is negative.

Remark 3

What about perpendicular?

CASE 1

One line is vertical (infinite slope). And the perpenduclar slope line is horizontal.

CASE 2

Given lines $L_1$, and line $L_2$, with slope $m_1$ and $m_2$, and $L_1$ is perpendicular to $L_2$, then

$$m_1=-\frac{1}{m_2}$$

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EXAMPLES

EXAMPLE 1

Given points $A\left(1,\:2\right)$ and $B\left(2,\:-1\right)$, Find the perpendicular slope of $\overline{AB}$.

EXAMPLE 2

Given points $A\left(3,\:1\right)$, $B\left(2,\:2\right)$, $C\left(0,\:1\right)$, and $D\left(1,\:0\right)$, do these points form a parallelogram?

EXAMPLE 3

Given points $A\left(-2,\:3\right)$, $B\left(0,\:2\right)$, and $C\left(2,\:0\right)$, Do these points lie on a straight line?

EXAMPLE 4

Given that $\Delta{ABC}$, $\Delta{ADC}$, and $\Delta{BDC}$ are all right similar triangles, determine whether the line $L_1$ and line $L_2$ are perpendicular in slope