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Slopes and Areas Frequently we will want to know the slope of a curve at some point. Or an area under a curve. We calculate area under a curve as the sum of areas of many rectangles under the curve. We calculate slope as the change in height of a curve during some small change in horizontal position: i.e. rise over run

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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 t y varies with x x varies with t Here we say “ y is a function of x”. Here we say “x is a function of t”.

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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.

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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 x- axis, a is the slope, and b is the place where the line hits the y-axis

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Straight Line Slope y = ax + b The slope, a, is just the rise y divided by the run x. We can do this anywhere on the line. So the slope of the line here is y = -3 x 2 Remember: Rise over Run and up and right are positive Proceed in the positive x direction for some number of units, and count the number of units up or down the y changes

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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

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We want an equation for this line y = ax + b is the general equation for a line So the equation of the line here is y = - 3 x + 3 2 Equation of our example line We plugged in the slope and y intercept

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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.

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Intersecting Lines The sum of angles on one side of a line equals 180 o P1 If angle AOB is 50 o, what is angle COD? P2 If angle AOB is 50 o, what is angle COB?

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Sum of angles in a Triangle The sum of angles in a triangle equals 180 o Notice this is a right triangle, because one of the angles (X0Y) is 90 o P3 if angle X0Y is 90 o, and angle 0XY is 60 o, what is angle 0YX?

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Review of Trig Sine = ord/hyp Cos = abs/hyp Tan = ord/abs P4 If = 60 o 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.

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Review of Trig Sine = ord/hyp (1) Cos = abs/hyp (2) Tan = ord/abs (3) P4 If = 60 o 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. Soln: Sine 60 o = 0.866 Solve Eqn (1) for ordinate ordinate = Sine * hypotenuse Plug in: ordinate = 0.866 * 2 meters ordinate = 1.732 meters

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