A cartoon showing a sad looking yellow point on a function, with a thought bubble that reads “Hello, I am point (x, f(x)). I want to find the slope exactly where I am, but the equation for slope requires 2 points.” The slope expression (change in y divided by the change in x) is written on the graph area.
A cartoon with two points on a function. The blue point that just showed up looks excited and is at some distance from the yellow point, with a speech bubble reading “I am your neighbor just delta x down the road, (x + delta x, f(x + delta x))! Can I help?” The yellow point responds “Well, let’s start by finding the average slope between us.”
The same two yellow and blue points separated by delta x on a function, with a green line representing the average slope between them. The equation for the average slope is show. Both points look satisfied.
Stylized text reading “So: the average slope between any two points on a function separated by delta x is quantity f(x+ delta x) - f(x) divided by delta x.”
The same two yellow and blue points separated by delta x, but now the yellow point is looking bummed out. The blue point says “What’s wrong?” and the yellow point responds “Well, I was hoping we could find my exact slope…”
Close up montage of the blue point, thinking hard about different ways to calculate an exact slope. In gray text behind them are equations for slope, tangents, delta x, crumpled papers. Finally in the third illustration in the montage, the blue point is holding their arms up in victory with a lightbulb above them and the word “LIMITS.”
Two cartoon points on a function. The blue point is swooping down toward the yellow point, yelling “I’M COMING OVER!” The yellow point looks kind of stunned and says “what.” The blue point responds “If I get really close to you, then our average slope will be close to your exact slope!” On the x-axis is gray text reading “delta x is shrinking!” as the two points move closer together.
Two cartoon points on a function, now very close together and with the blue point continuing to move toward the yellow point. In a speech bubble, the blue point says “And if the distance between us gets infinitely small (delta x approaches 0), our AVERAGE SLOPE becomes the INSTANTANEOUS SLOPE at a single point!” The yellow point now looks very happy.
A single green point, looking victorious with arms raised, which is now representing the yellow and blue points have converged (delta x is infinitely small). Text reads “The expression for the instantaneous slope at any point on a function, aka the derivative, is found by (1) Finding an expression for the slope between two points on the function separated by delta x, and (2) evaluating that slope as the points get infinitely close together.”