DISCOVERING THE CHAIN RULE THROUGH THE GRAPHING CALCULATOR

Transcription

1 DISCOVERING THE CHAIN RULE THROUGH THE GRAPHING CALCULATOR In this lab you will iscover the rule known at the chain rule for ifferentiation. This important rule will help you ifferentiate functions which are forme by the composition of or more functions. Exploration 1: Recall that if hx ( ) ( x 1), where h(x)=f(g(x)), then f( x) x an g(x)=x+1. At this point we o not have a ifferentiation rule for fining the erivative of h(x). Let's see if there is a relationship between f(x), f (x), g(x), an g'(x). Complete the chart below: x First graph the outsie function, f(x), using a thin line, an h(x), using a thick line, on the winow [-5,5]. Make a sketch of your graphs. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. Graphs of y1 an y. Table values for y1 an y What are the similarities an ifferences in these two functions f an h? What causes these ifferences an/or similarities? The function y is shifte 1 unit to the left of function y1. Graph f (x), using a thin line, an h'(x), using a thick line, in [-5,5]. Make a sketch of your graphs. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. The graphs of f an h. The table values for f an h.

2 ifferences an/or similarities? What is the equation which fits h'(x)? You may wish to put the gri on to help you write the equation for h'(x) or fin the erivative of h(x) by using expaning h(x) first an then factoring h'(x). The two graphs are parallel to each other which means they have the same slope. The graph of h is vertically shifte unit. It appears that f is x an h is x+ or (x+1). Complete the following chart using the information you have gathere. h(x)= (x+1) h'(x)= x+ or (x+1) Base upon the information liste, write what you observe to be a rule for fining h'(x) using the other functions: f(x), f (x), g(x), g (x). It appears to resemble the power rule. The exponent is brought own to the coefficient an the exponent is ecrease by 1. The insie function remains the same. Exploration : Let hx ( ) ( x 1) 3 Complete the chart below: f(x)= x 3 f (x)= 3x x. First graph the outsie function, f(x), using a thin line, an h(x), using a thick line, on the winow [-5,5]. Make a sketch of your graph. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. The graphs of y1 an y The table values for y1 an y. ifferences an/or similarities? Y is a horizontal shift of y1 to the left 1 unit. This can been seen in the graph an the table values. Graph f (x), using a thin line, an h'(x), using a thick line, in [-5,5]. Make a sketch of your graph. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1.

3 Graphs of y3 an y4. Table values for y3 an y4. ifferences an/or similarities? What is the equation which fits h'(x)? You may wish to put the gri on to help you write the equation for h'(x) or fin the erivative of h(x) by using expaning h(x) first an then factoring h'(x). The two graphs are shifts of each other. For example, the slope value at x = 1 on y3 is the same as the slope value on y4 at zero. This can been seen in both the graph an the table values. Y3 = 3x. Y4 appears to be 3(x+1). Which means that we are treating this as a power function. We are ifferentiating the power function an then placing the insie function in the results or h (x)=f (g(x)). Complete the following chart using the information you have gathere. f(x)= x 3 f (x)= 3x h(x)=(x+1) 3 h'(x)=3(x+1) Exploration 3: Let hx ( ) (3 x) Complete the chart below: g(x)= 3x g (x)= 3 x. First graph the outsie function, f(x), using a thin line, an h(x), using a thick line, on the winow [-5,5]. Make a sketch of your graph. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. Graphs of y1 an y. Table values for y1 an y.

4 ifferences an/or similarities? Both graphs are parabolic in shape, but y is a stretch of y1. The composition has change the steepness of most of the graph. They both pass through (0,0). Graph f (x), using a thin line, an h'(x), using a thick line, in [-5,5]. Make a sketch of your graph. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. Graphs of y3 an y4. Table values for y3 an y4. ifferences an/or similarities? What is the equation which fits h'(x)? You may wish to put the gri on to help you write the equation for h'(x) or fin the erivative of h(x) by using expaning h(x) first an then factoring h'(x). Both erivatives are linear. But the slope has change on y4 because the y was a stretch of y1. Using the table values it appears that the y4 = 6x which is 3(x). It appears that ifferentiating y4 involves bringing own the exponent an multiplying it by the erivative of f(x) an then lowering the power by 1. So h (x)= f (g(x))g (x). Complete the following chart using the information you have gathere. g(x)= 3x g (x)= 3 h(x)=(3x) Exploration 4: Let hx ( ) (x 3) h'(x)=3(x). Complete the chart below: 3 g(x)= x-3= x- g (x)= x. First graph the outsie function, f(x), using a thin line, an h(x), using a thick line, on the winow [-5,5]. Make a sketch of your graph. In aition make a table to

5 stuy both functions f an h. Set TblMin=-3 an tbl=1. The graphs of y1 an y. The table values of y1 an y. ifferences an/or similarities? Both graphs are parabolas. There has been both a horizontal shift an a stretch when y1 was change to y. The insie function part of x cause the vertical stretch an the -3 cause the horizontal shift. Graph f (x), using a thin line, an h'(x), using a thick line, in [-5,5]. Make a sketch of your graph. In aition make a table to stuy both functions f an h. Set TblMin=-3 an tbl=1. The graphs of y3 an y4. The tables for y3 an y4. ifferences an/or similarities? What is the equation which fits h'(x)? You may wish to put the gri on to help you write the equation for h'(x) or fin the erivative of h(x) by using expaning h(x) first an then factoring h'(x). The graph of y3 is simply y=x. The graph of y4 has a ifferent slope an a ifferent y- intercept. The slope of y4 appears to be 8 by looking at the table values. The y-intercept for y4 is -1.5 by looking at the table. So the equation of y4=8x-1=(4x-6)=(x-3)(). Putting this together with part III h (x)=f (g(x))(g (x)). Complete the following chart using the information you have gathere. 3 g(x)= x-3= x- h(x)= (x-3) g (x)= h'(x)=8x-1=(4x-6)=(x-3)() Extension: Use the rule you have evelope through the four explorations to fin the

6 erivative of each of the following functions. Begin by separating the composition of functions into the outsie function f(x) an the insie function g(x). 1. hx ( ) (3x 7) 4 h'( x) 4(3x 7) (3) 1(3x 7) 3. hx ( ) x h'( x) 3( x 10) ( ) ( x 10) h(x)=sin (10x) h'( x) cos(10 x) 10 10cos(10 x)