Transcript All the Calculus you need in one easy lesson

Slopes and Areas
• Frequently we will want to know the slope of
a curve at some point.
We calculate slope as the change in height of a
curve during some small change in horizontal
position: i.e. rise over run
• Or an area under a curve.
We calculate area under a
curve as the sum of areas
of many rectangles under
the curve.
Review: Axes
• When two things vary, it helps to draw a
picture with two perpendicular axes to show
what they do. Here are some examples:
y
x
x
y varies with x
Here we say “ y is a function of x” .
t
x varies with t
Here we say “x is a function of t” .
Positions
• We identify places with numbers on the axes
The axes are number lines that are perpendicular to each other.
Positive x to the right of the origin (x=0, y=0), positive y above the origin.
Straight Lines
• Sometimes we can write an equation for how one
variable varies with the other. For example a straight
line can be described as
y = ax + b
Here, y is a position on the
line along the y-axis, x is a
position on the line along the xaxis, a is the slope, and b is the
place where the line hits the y-axis
Straight Line Slope
y = ax + b The slope, a, is just the rise Dy divided by
the run Dx. We can do this anywhere on the line.
Proceed in the
positive x direction
for some number
of units, and count
the number of
units up or down
the y changes
So the slope of the line here
is Dy = -3
2
Dx
Remember: Rise over Run
and up and right are positive
y- intercept
y = ax + b is our equation for a line
b is the place where the line hits the y-axis
The intercept b is y = +3 when x = 0 for this line
y = ax + b is the general equation for a line
We want an equation for this line
Equation of our example line
So the equation of the line
here is
y = -3 x + 3
2
We plugged in the slope and y intercept
Intersecting Lines
• Intersecting lines make equal angles on
opposite sides of the intersection
• If a line intersects two parallel lines, equal
angles are formed at both intersections.
Intersecting Lines
• The sum of angles on one side of a line equals
180o
• P1 If angle AOB is 50o, what is angle COD?
• P2 If angle AOB is 50o, what is angle COB?
Sum of angles in a Triangle
• The sum of angles in a triangle equals 180o
•
Notice this is a right triangle, because
one of the angles (X0Y) is 90o
• P3 if angle X0Y is 90o, and angle 0XY is 60o,
what is angle 0YX?
Review of Trig
• Sine q = ord/hyp
• Cos q = abs/hyp
• Tan q = ord/abs
P4 If q = 60o and hyp = 2 meters
how long is the ordinate?
Hint: We know the hypotenuse and the
angle, so we can look up the sine. We
want the ordinate. The sine = ord/hyp, so
we can solve for the ordinate.
Review of Trig
• Sine q = ord/hyp (1)
• Cos q = abs/hyp (2)
• Tan q = ord/abs (3)
P4 If q = 60o and hyp = 2 meters
how long is the ordinate?
Soln: Sine 60o = 0.866
Solve Eqn (1) for ordinate
ordinate = Sine q * hypotenuse
Plug in:
ordinate = 0.866 * 2 meters
ordinate = 1.732 meters
Hint: We know the hypotenuse and the
angle, so we can look up the sine. We
want the ordinate. The sine = ord/hyp, so
we can solve for the ordinate.