Ay y f(x) P X

Suppose f is differentiable on an interval containing a, b, and let P(a,f(a)) and Q(b,f(b)) be distinct points on the graph of f.  Let c be the x-coordinate of teh point at which the lines tangent to the curve of P and Q intersect, assuming that the tangent lines are not parallel (see figure). Complete the following:

a. if f(x) = x^2, show that c = (a + b)/2, the arithemtic mean of a and b, for real numbers a and b.

The variable c is defined as the x-value of the point of intersection of the lines tangent to the points P and Q.

Compute the slope of line tangent to point P.

What is the equation of the line tangent to point P?

Compute the slope of the line tangent to point Q.

What is the equation of the line tangent to the point Q?

Find the point of intersection of the two lines.

b. if f(x)=sqrt(x), show that c = sqrt(ab), the geometric mean of a and b, for a > 0 and b > 0.

Compute the derivative of f(x)

Compute the slope of the line tangent to point P.

What is the equation of the line tangent to the point P?

Compute the slope of the line tangent to point Q.

What is the equation of the line tangent to the point Q?

Find the point of intersection of the two lines.

c = 

c. if f(x) = 1/x, show that c = 2ab/a+b, the harmonic mean of a and b, for a > 0 and b > 0.

Compute f`

f`(x) = 

Compute the slope of the line tangent to point P.

f`(a) = 

What is the equaiton of the line tangent to the point P?

y =

Compute the slope of the line tangent to point Q.

f`(b) = 

What is the equation of the line tangent to the point Q?

y = 

Find the point of intersection of the two lines.

c = 

d. Find an expression for c in terms of a and b for any (differentiable) function f whenever c exists.

To find the line tangent to a point, find the slope at the point which is f .

The slope of the line tangent to point P is

The equation of the line tangent to the point P is y =

The slope of the line tangent to point Q is

The equation of the line tangent to the point Q is y =

The x-value of intersection of the two lines is c =