1 Basic Differentiation Rules and Rates of Change Section 2.2

2 After this lesson, you should be able to: Find the derivative using the Constant Rule. Find the derivative using the Power Rule. Find the derivative using the Constant Multiple Rule and the Sum and Difference Rules. Find the derivative of sine and cosine. Use derivatives to find rates of change.

3 Basic Rules For Computing Derivatives The Constant Rule: The derivative of a constant function is zero. Examples: *The slope of a horizontal line is zero.

4 The Power Rule: If n is a rational number, then the function f(x) = x n is differentiable and Basic Rules For Computing Derivatives Examples:

5 The Constant Multiple Rule: If f is a differentiable function and, then cf is also differentiable. Basic Rules For Computing Derivatives Examples. Find the derivative of the function.

6 The Sum and Difference Rules: For f and g differentiable functions, Basic Rules For Computing Derivatives Examples 2) Find the derivative: 3) Find the slope of at (0, 1).

7 Basic Rules For Computing Derivatives Derivatives of sine and cosine Examples: FunctionDerivative

8 Finding the Equation of a Tangent Line Example: Find the equation of the tangent line to the graph of f at x = 1. Verify using calculator.

9 Finding an Equation of a Horizontal Tangent Line Example: Find an equation for the horizontal tangent line to the graph of

10 Rates of Change Rates of Change Used to determine the rate at which one variable changes with respect to another. For example,  velocity is the change in position w/ respect to time  acceleration is the change in velocity w/ respect to time  Water flow involves the change in height of the water w/ respect to time Average Velocity:

11 Velocity function, v(t): gives the instantaneous velocity of the object at time t. Motion Along a Straight Line Position function, s(t): gives the position of the object at time t relative to the origin. If s is positive, the object moved to the right (or upward) If s is negative, the object moved to the left (or downward) s = 0 is the starting position (origin) If v is positive, the object is moving forward or upward. If v is negative, the object is moving backward or downward. v = 0 means the object is stopped (at that very instant)

12 The speed of an object is the absolute value of the velocity. Velocity, Speed, and Acceleration Velocity is the rate of change of position with respect to time, thus we have v(t) = s’(t) * Measured in units of position over units of time. e.g. ft/s, m/s speed = |v(t)| Acceleration is the rate of change of velocity with respect to time, thus we have a(t) = v’(t) = s’’(t) * Measured in units of position over units of time squared. e.g. ft/s 2, m/s 2

13 Free Falling Object The position of a free falling object t seconds after its release can be represented by the equation For the position, measured in feet, of a free falling object, we have

14 Example of Free Falling Object #94 A ball is thrown straight down from the top of a 220-foot building with an initial velocity of –22 feet/second. a) What is the average velocity of the ball on the interval [1, 2]? b) What is its velocity after 3 seconds? c) What is its velocity after falling 108 feet? d) Find the velocity of the ball at impact First, we must write the position function that describes the motion of the object. Use as our model: In this example,

15 Example of Free Falling Object a) What is the average velocity of the ball on the interval [1, 2]? Use the fact that average velocity =

16 Example of Free Falling Object b) What is its velocity after 3 seconds? The velocity of the ball after 3 seconds can be expressed as v( ). Thus, we need an expression for the velocity function. Aha…now we need calculus!

17 Example of Free Falling Object (continued) c) What is its velocity after falling 108 feet? i) We need to find the time at which the object is ______ feet above the ground. That is, we need to find the value of t for which s(t) = _____. Solve Note: If the ball falls 108 ft, its position, s(t) = 220 – 108 = _______

18 Example of Free Falling Object (continued) c) What is its velocity after falling 108 feet? (continued) ii) Now we need to find the velocity at time t = _______. Verify your solution using the graphing calculator.

19 Example of Free Falling Object (continued) d) What is its velocity at impact? i)We need to find the time at which the object is ______ feet above the ground. That is, we need to find the value of t for which s(t) = _____. Solve Round to nearest tenth of second. ii) Now we need to find the velocity at time t = _______.

20 Homework Section 2.2 page 115 #1-59 odd, all, 93, 95, 103