Section 2.7 Derivatives and Rates of Change

Tangents Definition: The tangent line to the curve at the point is the line through P with slope provided that this limit exists.

Example 1 Find an equation of the tangent line to the parabola At the point

Example 2 Find an equation of the tangent line to the hyperbola at the point

Velocities Instantaneous velocity: Note:

Example 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. What is the velocity of the ball after 5 seconds? How fast is the ball traveling when it hits the ground?

Derivatives Definition The derivative of a function f at a number a, denoted by is if this limit exists.

Example 4 Find the derivative of the function

The tangent line to at is the line through whose slope is equal to ,the derivative of f at a. Point-Slope:

Example 5 Find an equation of the tangent line to the parabola at the point

Rates of Change Average Rate of Change of y with respect to x Instantaneous Rate of Change: The derivative is the instantaneous rate of change of with respect to x when

Example 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is dollars What is the meaning of the derivative What are its units? In practical terms, what does it mean to say that ? Which do you think is greater, or What about ? ?

Example 7 Let be the US national debt at time t. The table gives approximate values of this function by providing end of year estimates, in billions of dollars, from 1980 to 2000. Interpret and estimate the value of

Slope of the Tangent Line Derivative Instantaneous Rate of Change Velocity