MATH 1142 Section 1.1 Worksheet
NAME
One of the main concepts in calculus is that of the derivative
. The derivative is the instantaneous rate of change of a function. Sometimes calculating this rate is simple–as with a linear function–other times calculations need to be made using the average rate of change.
We have already briefly discussed linear functions.
Equations of Nonvertical Lines
A non vertical line L has an equation of the form y = mx + b where the number m is called the and the point (0 , b ) is called .
Because the average rate of change of a linear function is constant, the slope of the line is the derivative of the linear function .
Properties of the Slope of a Line:
Prop 1 If we start at a point on a line of slope m and move 1 unit to the right, then
.
Prop 2 We can compute the slope of a line knowing two points on the line, ( x
1
, y
1
) and ( x
2
, y
2
):
Prop 3 The equation of a line can be obtained using the slope m and one point ( x
1
, y
1
) on the line:
Prop 4 Distinct lines with the same slope are .
1

Prop 5 When two lines are perpendicular (excluding the case of vertical and horizontal lines),
.
Examples:
1. Find the slopes of the following lines:
(a) f ( x ) = 2 − 4 x
(b) y =
2 x − 1
3
(c) 6 x − 3 y = 1
2. Find an equation of the line lines:
(a) slope is − 1 / 3 and point on the line is (2 , 2)
(b) (1 , 2) and (3 , − 4) are on the line
(c) parallel to y = 6 and passes through the point (1 , 5)
(d) perpendicular to y = − 6 x + 1 and y -intercept is 0.
2