Second Derivatives and Graphs of Derivatives

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Presentation transcript:

Second Derivatives and Graphs of Derivatives AP Calculus Mrs. Agnew September 26-27, 2016

Essential Stuff Essential Question Essential Vocabulary How do you find second derivatives? How do you graph derivatives? Essential Vocabulary Second derivative Instantaneous Rate of Change Numerical Derivatives

Review: Tangent Lines How do you find the equation of a tangent line? What is the slope of the tangent line? What is the Calculus equation for the tangent line?

Review: Rates of Change How do you find the instantaneous velocity? What is the instantaneous rate of change? The derivative gives the slope of the tangent line, the instantaneous velocity and the instantaneous rate of change.

The Derivative Is… Slope of the curve Instantaneous Velocity Rate of Change Slope of the tangent line

Numerical Derivatives The numerical derivative is the value of the derivative at a given x value. Interpret the meaning of a numerical deriv. We can use the calculator to find the numerical derivative NDERIV dy/dx on graph screen Example: Use calc to find tangent line to y = xlnx at e and 1/e.

Second Derivatives The derivative of a function is a function itself… so it also has a derivative. The derivative of a derivative is known as the SECOND DERIVATIVE. Notations: How would you find the second derivative?

Second Derivatives Interpretation of the Second Derivative: “The rate of change of the rate of change” Rate of change of the slope of curve Acceleration In other words, it is how quickly something is increasing or decreasing. It is possible to find third, fourth, etc. derivatives as well. Guided Practice

Graphing Derivatives Key components of graph of derivative: When f is increasing, f '(x) > 0. When f is decreasing, f '(x) < 0. When f is at an extreme value, f '(x) = 0. Positive slope: derivative above x-axis. Negative slope: derivative below x-axis. Slope of zero: derivative on x-axis.

Graphing Derivatives Magnitude of slope tells you how far above and below the x-axis to go. Steeper slope means further away from x- axis. Stewart Page 178 #1, 3, 5, 6, 12, & Examples… Graphing Derivatives on Calculator

Homework: September 27 Stewart - page 167 – 170 #4, 5, 7, 10, 35, 37