SECTION 3.3: RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCESExample 1:3.1 GROUP EXERCISES 3.1 GROUP EXERCISES: DOING A LOT WITH A LITTLESection 3.1 introduced the Power Rule: ddxx nxn n 1, where n is a real number. The good news is that this rule, combined with the Constant Multiple and Sum Rules, allow us to take the derivative of even the most formidable polynomial with ease! To demonstrate this power, try Problem 1:1. A formidable polynomial: f x x x x x x x x( ) .       10 9 8 7 6 5 479125 0 33 2 42Its derivative:f x( )The ability to differentiate polynomials is only one of the things we’ve gained by establishing the Power Rule. Using some basic definitions and a touch of algebra, there are all kinds of functions that can be differentiated by using the Power Rule.2. All kinds of functions:f x x( )  3 52g xxx( )  1 133433 2( )x xh xx+ +=Their derivatives:f x( )g x( )h x( )Unfortunately, there are some deceptive functions that look like they should be straightforward applications of the Power and Constant Multiple Rules but actually require a little thought.3. Some deceptive functions:f x x( )  24bgg x x( ) 35chTheir derivatives:f x( )g x( )433.1 COMPUTER EXAMPLES SECTION 3.1: DERIVATIVES OF POLYNOMIAL AND EXPONENTIALFUNCTIONSExample: At what point(s) on the graph of y x x x   3 23 5 6 is the tangent line parallel to the line y x4?To begin, a graph would be helpful. In Converge, choose Graph, then Graph without a file, choose Y = F(X) and 2 functions. Type the first function x^3  3x^2  5x + 6, hit the TAB key, type the second function 4x and hit ENTER. Use this window: Min X: 8, Max X: 8, Min Y: 15, Max Y: 12.  Looking at the graph, how many points will there be where the tangent line is parallel to the line y x4? What are the approximate locations of these points?To find these points, you need the derivative, since it represents the slope of the curve. Find the derivative of the polynomial. What value should the derivative have at these points? Write and solve an equation to find these x-values.443.1 COMPUTER EXAMPLES  To test your answers, write the equations of the tangent lines at these points and then graph them. To graph each line, choose Post-graph, Overlay any type of graph, Y = F(X), and type theequation 453.3 CLASS EXAMPLES SECTION 3.3: RATES OF CHANGE IN THE NATURAL AND SOCIALSCIENCESExample 1: Given a function ( )y f x, yx gives the average rate of change of y withrespect to x over some interval, whereas the derivative dydx can be interpreted as the instantaneous rate of change of y with respect to x. For example, if ( )s f t gives the height (in meters) of a vertically-projected object at time t seconds, then:a.(3) 20f  means that at 3 seconds, the object has a height of 20 meters.b.st gives the average rate of change of height compared to time, or the average velocity over a time interval. If we know that st= 35, then the object has an average velocity of 35 meters per second over the measured time interval. Note that the units of st are meters per second.c.( )f t gives the instantaneous rate of change of height compared to time, or the instantaneous velocity at time t. In this example, ( )f t will be positive as the object rises and will be negative when the object is coming down.d.(3) 20f means that at 3 seconds, the object is rising (since the derivative is positive) at the rate of 20 meters per second. The units of the derivative are in meters per second.e.(6) 35f means that at 6 seconds the object is falling at the rate of 35 meters per second.Example 2: The temperature, T, in degrees Fahrenheit, of a cold potato placed in a hot oven is given by ( )T f t, where t is the time in minutes since the potato was put in the oven.a. What is the practical meaning of the statement (5) 80f ?b. What is the sign of ( )f t? Why?c. What are the units of (20)f? What is the practical meaning of the statement(20) 2f?463.3 CLASS EXAMPLES Example 3: You may be interested in determining how the price of the item your business sells affects its sales. Suppose that at a price of $p, a quantity, q, of the item is sold and the quantity function is ( )q f p.a. What is the practical meaning of the statement (20) 100,000f ?b. What is the practical meaning of the statement (20) 2000f? How might this be used to predict sales if the price is $21?Example 4: Suppose ( )g v is a function that gives the number of miles per gallon a car gets (fuel efficiency) when it is going v miles per hour.a. What is the practical meaning of the statement(45) 20g ?b. What are the units of (55)g? What is the practical meaning of the statement(55) 0.54g?Example 5: A ball is thrown vertically upward with a velocity of 40 ft/sec. Its height after t seconds is 2( ) 80 16s t t t= -.a. Find the velocity, v(t), of the ball at time tb. Find (1)vand explain its practical meaning in this situation. What does the sign of (1)vsignify? c. Find (3)vand explain its practical meaning. What does the sign of (3)vsignify? 473.3 CLASS EXAMPLES d. Find (2.5)vand explain its practical meaning in this situation. What does the value of (2.5)vsignify here? e. Find the acceleration, a(t), of the ball at time t. Explain why this function has a negative sign? f. Think about what happens when a ball is thrown vertically upward. When is the ball speeding up and when is it slowing down?g. Use the velocity and acceleration functions to fill in the following table.Time interval (sec.) Sign of Velocity Sign of Accelerationt = 1t = 2t = 2.5t = 3t = 4Conclusion:  When the ball is slowing down, the sign of velocity is _____________ and the sign of acceleration is ______________ If the ball is slowing down, the velocity and acceleration have ________________ signs When the ball is speeding up, the sign of velocity is ________________ and the sign ofacceleration is ________________ If the ball is speeding up, the velocity and acceleration have ________________ signs483.5 COMPUTER EXAMPLES SECTION 3.5: THE CHAIN RULEExample: Investigate the derivative of f x x( ) sin( ) 2.Because the derivative of f x x( ) sin is f x x( ) cos, you might think that the derivative of sin(2x) is cos(2x). Let’s see