Connecting f, f', and f'' graphically | AP Calculus AB | Khan Academy

TL;DR

Analyzing three graphs of functions and their derivatives to determine which graph represents the original function, first derivative, and second derivative.

Transcript

• [Instructor] We have the graphs of three functions here, and what we know is that one of them is the function f, another is the first derivative of f, and then the third is the second derivative of f. And our goal is to figure out which function is which. Which one is f, which is the first derivative, and which is the second? Like always, pause t... Read More

Key Insights

• 💁 Analyzing the slopes of graphs can provide information about the behavior of functions and their derivatives.
• 📈 The derivative of a function should have a trend opposite to that of the original function.
• 👈 The intersection of a graph with the x-axis can indicate key points for determining the derivative.

Summary & Key Takeaways

• The first graph (orange) represents the original function, starting with a positive slope, crossing the x-axis with a zero slope, and then becoming increasingly negative.

• The second graph (blue) does not represent the derivative of the first graph as its trend is opposite, going from negative to positive.

• The third graph (magenta) has the correct trend, intersecting the x-axis at the right point, and is positive when the tangent line slope is positive, making it a potential candidate for the first derivative.