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Summary AP Calculus AB/BC: Understanding the limit definition of the derivative. $9.39   Add to cart
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Summary AP Calculus AB/BC: Understanding the limit definition of the derivative.
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  • Course
  • AP Calculus AB
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  • Senior / 12th Grade

This is an in-depth comprehensive review of the definition of the derivative, the limit definition, how to go from a finding the rate of change over an interval to finding the rate of change of a function at an instant, and AP exam tips to help you get a 5/5 or perfect score!

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  • November 6, 2024
  • 9
  • 2024/2025
  • Summary

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  • Senior / 12th grade
  • AP Calculus AB
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Limit Definition Of Derivative
(studying tips for the AP exam are at the bottom)
Suppose we have a straight line given by the equation f(x)=2x as shown in the graph below




We know we can find the rate of change or “slope” of the line at any point by using the rise/run
formula which is

∆𝑦 𝑦2−𝑦1
∆𝑥
= 𝑥2−𝑥1
∆𝑦 represents the change in the y-value AKA “rise” or how much the graph goes up on the
interval, and ∆𝑥 represents the change in the x-value AKA “run” or how much the graph moves
horizontally on the interval. The most important par tof this is that for a straight line, this rise
over run, or slope, is the same for every x-value.

2
But what about a curvy function like 𝑓(𝑥) = 𝑥 as shown in the graph below?

, You can try to figure out the average rate of change on any interval, but since the average rate of
change on any interval is different because the function gets steeper as you move to higher/lower
x-values, you could never get the “slope” of the function for every x-value with this method.

Well, as we know the average rate of change of a function is given by

∆𝑦 𝑦2−𝑦1
∆𝑥
= 𝑥2−𝑥1
Let’s replace the “y” with f(x) to better show function notation

∆𝑓(𝑥) 𝑓(𝑥2)−𝑓(𝑥1)
∆𝑥
= 𝑥2−𝑥1
Instead of representing the two points using 𝑥1 and 𝑥2, let’s represent them differently.
Let 𝑥1 stay as only “x” and let 𝑥2 be a small distance distance “h” away from 𝑥1, meaning we can
represent it as just “x+h” as shown in the graph below.

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