Slope of the secant line? PLEASE HELP

Question

the point P(25,9) lies on the curve y= √x+4. Let Q be the point (x, √x+4)

 

 

A) Find the slop of the secant line PQ for the following values of x (Answers here should be correct to at least 6 places after the decimal point.):

 

x= 25.1, slope of PQ= ?

x=25.01, slope of PQ = ?

x=24.9 , slope of PQ = ?

x=24.99, slope of PQ = ?

 

 

B.) Based on the above results, estimate the slope of the tangent line to the curve at P(25 , 9 ).

 

 

 

Answer 1

For each value of x, find the value of y, using the equation of the curve (y= √x+4).  That's easy to do on a calculator.

 

This will give you the coordinates of the point Q for each value of x.

 

Then use the slope formula to calculate the slope of PQ.

 

For example, if x = 25.1, then y = 9.01.

The slope of the line from (25, 9) to (25.1, 9.01) is (9.01 - 9) / (25.1 - 25) = .01 / .1 = 0.1

 

Do that for the other three values of x that they give, and you will have all of part A done.

 

For part B, remember that as you get closer to the point in question (25,9), the secant lines get closer to being the actual tangent to the curve at that point.  So from the values in Part A, can you see a pattern?  As you get closer to (25, 9), does the slope of the secant seem to be approaching some specific value, that you could guess is a good estimate for the slope of the tangent at that point?