2 parts. help plz. show work

Part (a)

Use the slope formula to compute the slope from x = 4 to x = 6

So effectively we're finding the slope of the line through (4,70) and (6,68)

We get the following

m = (y2-y1)/(x2-x1)

m = (68-70)/(6-4)

m = -2/2

m = -1

Repeat for the points that correspond to x = 6 and x = 8

m = (y2-y1)/(x2-x1)

m = (73-68)/(8-6)

m = 5/2

m = 2.5

Now average the two slope values

We'll add up the results and divide by 2

(-1+2.5)/2 = 1.5/2 = 0.75

The estimate of T'(6) is 0.75

This works because T'(x) measures the slope of the tangent line on the T(x) curve. Averaging the secant slopes near x = 6 will help give us an estimate of T'(6), which is the slope of the tangent at x = 6 on T(x).

Part (b)

The value T'(6) = 0.75 represents the instantaneous rate of change of the temperature per hour.

More specifically, T'(6) = 0.75 means the temperature is increasing by an estimated 0.75 degrees per hour at the exact instant of x = 6 hours. This instantaneous rate of change is like a snapshot at this very moment in time; in contrast, the slope formula results we computed above measure the average rate of change between the endpoints mentioned.

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