1. Using Maple to find the equation of a tangent line. .

Recall that if ( ) is a point on a line with slope m , then an equation of the line is given by , or if we want to express it as a function in the variable x , we could define the line by , where . If we want to find the equation of the tangent line to a function at the point ( ), we first need to compute the slope, which is given as the limit of the difference quotient: . To illustrate this idea, suppose that we define

> f:=x->sin(x)+cos(x);

> x0:=Pi;

Then the difference quotient is given by

> dq:=h->(f(x0+h)-f(x0))/h;

> dq(h);

Let us use maple to compute .

> m:=limit((-sin(h)-cos(h)+1)/h,h=0);

Let us define the function whose graph is the tangent line.

> L:=x-> m*(x-x0)+f(x0);

> L(x);

Finally, let us plot the function and its tangent line on the same graph.

> plot([L,f],0..2*Pi,color=[red,blue]);

Notice that the line appears to be tangent to the curve at the point ( ).

Submission:

For the following functions:

(a) Plot over the interval [ ] .

(b) Define the function with Maple.

(c) Find .

(d) Define the secant lines for and 1. Graph them together with and the tangent line over the interval in part (a).

1. ,

2. ,

3. ,

4. ,

Submission worksheet: