Tangent Lines and Derivatives

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Presentation transcript:

Tangent Lines and Derivatives AP Calculus September 19 & 20, 2016 Mrs. Agnew

Essential Question Essential Vocabulary What is the derivative and how do you calculate it? Essential Vocabulary Slope of a curve Instantaneous rate of change Tangent line Derivative Numerical Derivative

Tangent Lines and Velocities When computing the slope of a tangent line, why is it necessary to evaluate a limit? The formula above gives us the slope of the tangent line to the curve at x = a. Also gives the instantaneous velocity of an object at time t = a. Instantaneous Rate of Change

The Derivative THE DERIVATIVE The special limits below occur so widely, they are given a special name… THE DERIVATIVE Examples…

What is the Derivative? The derivative represents… the slope of the tangent line the instantaneous rate of change the instantaneous velocity If the derivative is slope of tangent line, then equation of tangent line at x = a is: Examples…

What is the Derivative? The instantaneous rate of change is the limit of the average rates of change over smaller and smaller intervals. The derivative, f ’(a), is the instantaneous rate of change in y = f(x) with respect to x at x = a. Examples…

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

Homework: 9/20/2016 Page 155 – 156 (Stewart) #3 – 7, 10, 13, 14, 15, 18, 25, 28, 30, 33