How Secant Lines become Tangent Lines Adrienne Samantha

How Secant Lines become Tangent Lines

Adrienne, Samantha, Danielle, and Eugene, trying to raise Walkathon money, build a 196 foot high-dive platform in the middle of the plaza. After charging admission for prime bleacher seats in the plaza, they then “persuade” Mr Murphy to be the first high diver. Abby, Nica, and Hamidou observe Mr Murphy’s “dive” closely enough to form an equation for his height. They find the equation to be given by: …and the graph is given by…

t = Time in seconds s(t) = Height off of the ground(in feet)

(0, 196) because at time = 0, Mr. Murphy is at the top of the 196 foot high platform. What is Mr. Murphy’s average velocity during his 3. 5 second plunge? s(t) = Height off of the ground(in feet) What are the coordinates for this point? t = Time in seconds At what time did Mr. Murphy land? (3. 5, 0) which you can find by setting s(t) = 0 Why is the velocity negative? = − 56 feet/sec Because the motion is downward.

How can you represent the average velocity on this graph? s(t) = Height off of the ground(in feet) Since you are all experts at algebra… t = Time in seconds = 56 feet/sec …can be shown graphically to be… The average velocity can be represented as the slope of the secant line through the initial and final points. from time 0 to time 3. 5

s(t) = Height off of the ground(in feet) Find Mr. Murphy’s average velocity between 1 and 3 seconds. t = Time in seconds = 64 feet/sec …the slope of the line through the initial and final points. from time 1 to time 3

We can draw a secant line close to 3. we’ll start with 2 seconds. s(t) = Height off of the ground(in feet) Approximate Mr. Murphy’s instantaneous (exact) velocity at 3 seconds. t = Time in seconds …which is close to the exact velocity at 3 seconds. = 80 feet/sec from time 2 to time 3

We can even try a secant line through 2. 5 and 3. s(t) = Height off of the ground(in feet) Approximate Mr. Murphy’s instantaneous (exact) velocity at 3 seconds. t = Time in seconds …which is even closer to the exact velocity at 3 seconds. = 88 feet/sec from time 2. 5 to time 3

We would need to get the secant points as close as we can. How would we do that? By taking the LIMIT as one point approaches the other… s(t) = Height off of the ground(in feet) I think we’re getting the idea of how to find Mr. Murphy’s instantaneous (exact) velocity at 3 seconds. t = Time in seconds

We would need to get the secant points as close as we can. How would we do that? By taking the LIMIT as one point approaches the other… s(t) = Height off of the ground(in feet) I think we’re getting the idea of how to find Mr. Murphy’s instantaneous (exact) velocity at 3 seconds. t = Time in seconds In other words…

So from this we find that as the two points on a secant line approach each other, it becomes the tangent line. And the slope of the tangent line is also called… The Derivative Hey! This is from …which can also be Algebra class! written as… Let And since

f(x) x p