[size=150]This is a visualization of the product rule in calculus for functions that are the product of two functions. [br][br]For example, [math]f\left(x\right)=x\sqrt{x}[/math]. This is a product of [math]u\left(x\right)=x[/math] and [math]v\left(x\right)=\sqrt{x}[/math];[/size][size=150][br][br]We are going to look at a function, [math]f\left(t\right)=u\left(t\right)v\left(t\right)[/math] The basic idea is that [b]the product of two functions can be visualized as the area of a rectangle.[/b] [br][br][b]NOTE:[/b] the curve below is [b]PARAMETRIC[/b], meaning that each point on the curve represents (u(t), v(t)) for some t, NOT the point (t, f(t)). [br][/size]

[size=150]In the applet [b][color=#980000]ABOVE[/color][/b][br][br][list][*][b]SET h = 0;[/b] we will define the rectangle you see as our INITIAL rectangle.[br][/*][/list][list][*][b]SET h = 0.6;[/b] calculate the INCREASE in the area by calculating the sum of the areas of the blue, green and brown rectangles. Then, divide this area by the value of [i]h[/i], giving your answer correct to [color=#ff0000]3-significant figures[/color].[/*][/list][list][*][b]SET h = 0.4; [/b]repeat what you did above.[/*][/list][list][*][b]SET h = 0.2;[/b] repeat what you did above.[br][/*][/list][list][*][b]SET h = 0.1;[/b] repeat what you did above.[br][br]What do you notice about the values you calculated as you reduced the value of h?[/*][/list][br][/size]

[size=150]Consider the expression shown in the applet [color=#980000][b]BELOW[/b][/color].[br][br]What does [b]NUMERATOR[/b] represent?[br][br][b]Set the value of [i]t[/i] to [i]t [/i]= 2. [/b]As you move the value of [i]h[/i], closer to 0 what does the value of the fraction represent?[br][br][b]Set the value of [i]t[/i] to [i]t[/i] = 1.[/b] Again as the value of [i]h[/i] moves closer to 0 what happens to the value of the fraction?[br][br][/size][size=150]What do the values you have found represent?[/size]

[size=150][br]The [b][color=#980000]PRODUCT RULE [/color][/b]says that;[br][br][math]\frac{d\left(u\left(t\right)\times v\left(t\right)\right)}{dt}=v\times\frac{d\left(u\left(t\right)\right)}{dt}+u\times\frac{d\left(v\left(t\right)\right)}{dt}=v\times u'+u\times v'[/math][br][br]Show how to use the expression in the applet ABOVE to derive this rule.[br][br][b]HINT 1: [/b]split the fraction up into THREE separate fractions...then go from there![br][br][b]HINT 2:[/b] Recall that; [math]lim_{h\longrightarrow0}\frac{f\left(x+h\right)-f\left(x\right)}{h}=f'\left(x\right)[/math][/size]