How to Find Concavity in Calculus : Calculus Explained

TL;DR

Learn how to determine concavity in calculus by analyzing the sign of the second derivative.

Transcript

hi there this is Ryan Malloy here at the worldwide centre of mathematics in this video we're going to discuss how to find concavity in calculus so before we begin let's go over what concavity means in a visual sense when something is concave down it takes this general shape where as it moves to the left it goes down and move to the right and goes d... Read More

Key Insights

• 📈 Concavity in calculus refers to the shape of a curve, with concave down resembling a downward U and concave up resembling an upward U.
• 🤘 The sign of the second derivative determines the concavity of a function.
• 😥 To find concavity, the second derivative is set equal to zero to find critical points.
• 😥 Test values are used to determine the concavity on either side of the critical points.
• ❓ The concavity of a function can be concave down, concave up, or neither.
• ♾️ In this example, the function is concave down from negative infinity to -1/3 and concave up from -1/3 to positive infinity.
• 😥 The concavity of a function affects the behavior of its graph, such as the presence of inflection points.

Questions & Answers

Q: What is concavity in calculus?

Concavity in calculus refers to the shape of a curve. A curve is concave down if it moves downwards as you go from left to right, and concave up if it moves upwards in the same direction.

Q: How is concavity determined?

Concavity is determined by analyzing the sign of the second derivative of a function. If the second derivative is positive, the function is concave up. If it is negative, the function is concave down.

Q: What is the process for finding concavity?

The first step is finding the first derivative of the function. Then, find the second derivative. Next, set the second derivative equal to zero to determine the critical points. Finally, test values on either side of the critical points to determine the concavity.

Q: How are test values used to determine concavity?

Test values are used to evaluate the sign of the second derivative at different points. If the result is positive, the function is concave up in that region. If it is negative, the function is concave down.

Summary & Key Takeaways

• Concavity refers to the shape of a curve, with concave down resembling an upside-down U and concave up resembling a U.

• To find where a function is concave up or down, the sign of the second derivative is analyzed.

• By setting the second derivative equal to zero and testing values on either side, the concavity of the function can be determined.