Calculus Survival Facts
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4Calculus Survival Facts: Algebra
Eric Key
Abstract
This is the bare minimum of algebra you need to be successful in the first semester of calculus.
1
Distance and Midpoint
1. The absolute value of x, denoted by x is the distance from x to 0.
2. x
y is the distance from x to y.
2
2
3. The distance from (a, b) to (c, d) is
a
c
+ b
d
.
2
2
2
4. The circle with radius R and center (a, b) has an equation of the form (x a)
+(y b)
= R
.
Conversely, each such equation describes a circle of radius R and center (a, b).
5. If 0
t
1 and (a, b) and (c, d) are given, then the point
((1
t)a + tc, (1
t)b + td)
lies t of the way from (a, b) to (c, d). If t = 1/2 this is the midpoint of the line segment
joining (a, b) and (c, d).
2
Linear Equations
1. A line is determined by 2 distinct points.
2. Every line has an equation of the form Ax + By + C = 0. This is the general equation of
2
2
a line. Conversely, if A
+ B
= 0 then Ax + By + C = 0 is the equation of a line. This
equation is called the general equation of a line.
3. The distance from the point (a, b) to the line with equation Ax + By + C = 0 is given by
Aa + Bb + C
.
2
2
A
+ B
4. Every line parallel to the line Ax+By+C = 0 has an equation of the form Ax+By+D = 0.
5. Every line perpendicular to the line Ax + By + C = 0 has an equation of the form Bx
Ay + D = 0.
6. If a line passes through the points (a, b) and (c, d) and a = c then it has an equation of the
form
x
c
x
a
y = b
+ d
.
a
c
c
a
This is the interpolation equation of the line.
7. If a line passes through the points (a, b) and (c, d) and a = c then the quantity
d
b
c
a
is called the slope of the line. Every line has at most one slope. A line is said to be
vertical if its slope cannot by defined, and horizontal if its slope is 0.
8. If two lines are given and neither is vertical, then they are perpendicular if and only if the
product of their slopes is
1.
9. If the slope of a line is m and it passes through (a, b) then y
b = m(x
a) is an equation
of this line, called the point-slope form of the equation. Conversely, y
b = m(x
a) is
an equation of a line with slope m and passing through (a, b).
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