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## Applied Calculus (MAT 121) Dr. Day Wednes day Feb 22, 2012

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**Applied Calculus (MAT 121)Dr. Day Wednesday Feb 22, 2012**• Using the Second Derivative • Is a function’s first derivative increasing or decreasing? • Inflection points and concavity • Increasing at an increasing rate • Increasing at a decreasing rate • Decreasing at an increasing rate • Decreasing at a decreasing rate MAT 121**What does the second derivative tell us?**• Not only can a function increase or decrease, it can do so in various ways. The second derivative helps us identify patterns in how a function increases or decreases. These patterns help us describe the concavity of a function. • The key points where concavity changes are called points of inflection. The second derivative helps us identify these key points. • Specific contextual situations are more thoroughly analyzed using the second derivative. Connections among position, velocity, and acceleration can be made using first and second derivatives. MAT 121**What Possible Ways Can a Function Increase or Decrease?**MAT 121**What Possible Ways Can a Function Increase or Decrease?**MAT 121**The Second Derivative Reveals a Function’s Concavity**Example: Determine the intervals on which the function f(x) = x3 – 3x2 – 24x + 32 is concave up and on which it is concave down. Identify all points of inflection. MAT 121**The Second Derivative Reveals a Function’s Concavity**Example: Determine the intervals on which the function f(x) = x+ 1/xis concave up and on which it is concave down. Identify all points of inflection. MAT 121**Assignments**WebAssign 4.1 due tonight. 4.2 due Thursday 2/23. Reminder: Check results for Test #2 MAT 121