• First, define a function f with the formula f(x) = sin(x) + cos(x).
• Next, define a function g so that the graph of g is a straight line tangent to the graph of f at the point (p/2, 1).
• Finally, go to the Y= window and define y1(x)=f(x) and y2(x)=g(x). On the GRAPH screen you can then see the graph of f and the tangent line to that graph at (p/2, 1).

 

How is the new curve related to the old one?

How is the new tangent line related to the old one?

How are the slopes of the tangent lines related?

How are the points of tangency related?
 

2. Go to the Y= screen and define y5(x)=f(x+2) and y6(x)=g(x+2). Look at the graphs.

Answer the same questions as in #1.

(It will probably help if you delete the graphs of y3 and y4 ; highlight them on the Y= screen and press F4 to remove the checkmark. The formulas are still there, but the calculator won't draw the graphs.)
 

3. Go to the Y= screen and define y7(x)=2f(x) and y8(x)=2g(x). Look at the graphs.

Answer the same questions as in #1.
 

4. Go to the Y= screem and define y9(x)=f(2x) and y10(x)=g(2x). Look at the graphs.

Answer the same questions as in #1.
 

5. Now try more complicated constructions, like f(2x+1) or 2f(x)+1. Note that if h(x) = 2x+1, these are just f(h(x)) and h(f(x)). Look at the graphs.

Answer the same questions as in #1.

If you're having trouble answering the questions by looking at the graphs, go back to the HOME screen and type g(x)+2, etc. on the command line. The calculator will give you the formulas for the revised functions; you may need the expand command to get the formula you want.
 

Now try a different function, like f(x) = e^x at the point (-1, e^-1). All you need to do is change the definitions of f(x) and g(x) on the HOME screen; you already have the right formulas on the Y= screen.
 

Generalize: If you know f'(a) for some function f and some number a, what do you know about the derivative of f(3x+2)? (What value does it have at what input?)

What do you know about the derivative of f(rx+s) for non-zero numbers r and s?

If h is a differentiable function and h(b) = a and h'(a) = m, what do you know about the derivative f(h(x))? (If it helps, assume h is linear.)
 

You are now an expert on the chain rule of differentiation.

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