Please Help me!Finding derivatives base off of a data set:

Question

A function T(x) is continuous and differentiable with values given in the table below. Use the values in the table to estimate each of the following. Use best estimates and round your answers to two decimal places

x       45 48 51 54 57
T (x) 25 17 15 32 28

 

(b) The rate of change of T (x) at x = 51 is approximately

For this I thought you would divide15/51, and I got .29 but that was wrong so I don't know how to do this one.

(c) lim (T(48 + h) − T(48))/h
    h→0 

For this I first plugged in ) for h and got zero but that was wrong. So then I started plugging in numbers extremely close to zero and I always got 1 but that was wrong as well. Please explain what I need to do to complete this!

Thank you!

Answer 1

First (I cannot resist this, remembering an English teacher from long ago): NEVER write or say "OFF OF".

 

x        45 48 51 54 57
T (x) 25 17 15 32 28

 

The table has constant difference of 3 in its successive x values,  i.e, Δx is 3.

 

To estimate the derivative (slope) when x=51, you can use Δy = 17, i.e the difference between the y(54) and y(51). So that gives slope 17/3.

 

Alternatively, you could use x=48 as the start & x = 51 as the end, in which caseyou'd get Δy=-2 and slope -2/3.

 

Or you could average these two slopes. All in all, a fairly useless exercise.  

 

Perhaps better would be to plot these points, do an appropriate regression, and then differentiate the fitted y function, evaluating it at x=51.

 

Your attempt, which seems to have been just the ratio of y to x, is never a correct approach.  

 

I don't see how to do c) unless you have a defined function for T(x).