Chapter 3: Derivatives 3.1 Derivatives and Rate of Change

Differential Calculus Study of how one quantity changes in relation to another quantity Central concept is the derivative – Uses the velocities and slopes of tangent lines from chapter 2

Derivatives Special type of limit Like what we used to find the slope of a tangent line to a curve Or finding the instantaneous velocity of an object Interpreted as a rate of change

Slope of a Tangent Line If a curve has an equation of y = f(x), and we want to find the slope of a tangent line at some point P, then we would consider some nearby point Q and compute the slope of that “tangent line” as:

Slope of a Tangent Line (cont.) Then, we would pick a point Q that is even closer to P, and then closer, and closer The number that the slope would approach as we got closer to P (the limit) was the slope of the tangent line

Formal Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope: As long as the limit exists!

Example 1 Find an equation of the tangent line to the parabola y = x 2 at the point P(1,1).

Another definition for the slope of a tangent line… As x approaches a, h approaches 0 (because h = x – a) So, the slope of the tangent line becomes the equation above!

Example 2 Find an equation of the tangent line to the hyperbola y = 3/x at the point (3,1).

Average Velocity Ave velocity = displacement time

Instantaneous Velocity we take the average velocity over smaller and smaller intervals (as h approaches 0) Velocity v(a) is the limit of the average velocities:

Instantaneous Velocity cont. This means that the velocity at time t = a is equal to the slope of the tangent line at P. Let’s reconsider the problem of the falling ball…

Example 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. (a) what is the velocity of the ball after 5 seconds? Remember the equation of motion s = f(t) = 4.9t 2

Example 3 cont. Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. (b) how fast is the ball traveling when it hits the ground?

Definition The derivative of a function f at a number a, denoted by f’(a), is: If that limit exists.

Equivalent way to write the derivative

Example 4 Find the derivative of the function f(x) = x 2 – 8x + 9 at the number a.

A note…. The tangent line to y = f(x) at some point (a,f(a)), is the line through (a,f(a)) whose slope is equal to f’(a), the derivative of f at a.

Example 5 Find an equation of the tangent line to the parabola y = x 2 – 8x + 9 at the point (3,-6).

Rates of Change Suppose y is a quantity that depends on another quantity x This means y is a function of x, and we write y = f(x) If x changes from x 1 to x 2, then the change in x (called the increment of x) is: The corresponding change in y is:

Rates of Change (cont.) The difference quotient (slope) is: Called the average rate of change of y with respect to x

Instantaneous Rate of Change

Definition The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

Homework P , 7, 9 a&b, 13, 17, 25, 27, 29