Calculus 220 section 1 1 The Slope of a Line notes by Tim Pilachowski Bring your graphing calculator to the next Lecture You might ask Why are we talking about lines again I did this in high school and again in Math 113 Haven t we said everything there is to say The answer lies in knowing that it is not the lines themselves which are of interest but rather it is their slopes Slope is a measurement of change and Calculus is the mathematics of change If we can look at the rate of change in a situation determine whether the rate of change is constant or whether it fluctuates then we can use those observations to find a mathematical model i e equation to describe the situation The model can then be used not only to evaluate the past but also to predict and plan for the future Examples A B Consider the following two graphs of linear equations x y 5 and y 5 x 3 By defining slope as change in y y we address both direction and steepness Answers 1 5 change in x x Example C slope of a horizontal line and slope of a vertical line Answers 0 undefined You ll also need to know about equations of horizontal and vertical lines Examples A B extended slope intercept form of the equation f x mx b Example D 3 x 2 y 7 Answer 32 Example E If a new car costs 24000 and its value depreciates goes down by 2000 each year Find a mathematical model i e equation Answer 2x 24 Example E extended If you only have 10000 to spend on a used car about how old will it be to still be in your price range Answer 7 years Example F Find the equation of the line which passes through the points 5 3 and 10 0 Answer y 35 x 6 Example G When a company spent 10000 on radio ads they had revenue of 8 500 000 When they spent 20000 their revenue rose to 9 million Find the mathematical model for revenue as a function of amount spent on advertising Answer 0 05x 8 Example H Find an equation to relate C to F Answer F 9 C 32 or C 5 F 32 5 9 Example I Parallel lines will necessarily have equal slopes the over and up or over and down movements must match since the two lines go in the same direction You learned that perpendicular lines have slopes that are negative reciprocals Must this be true Consider the line y 2x and its perpendicular With a little basic geometry and the knowledge that the three angles of a triangle must add up to 180 it s not too 2 hard to show that the over 1 up 2 triangle of y 2 x is transposed into an over 2 down 1 triangle for the perpendicular line So while the slope of y 2x is 1 2 1 1 2 m 2 the slope of the perpendicular must be m the negative 1 1 2 2 reciprocal The same principles can be used to prove the relationship is true for any perpendicular lines
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