# Average Rate of Change Using MVT for Derivatives

#### carlodelmundo

1. Homework Statement

The mass, m(t), in grams, of a tumor t weeks after it begins growing is given by m(t) = [te^t] / 80 .

What is the average rate of change, in grams per week, during the fifth week of growth?

a.) 2.730
b.) 3.412
c.) 6.189
d.) 6.546
e.) 11.131

2. Homework Equations

The Mean Value Theorem (MVT) for Derivatives states that the average rate of change between two points is the secant line between those two points given by the equation:

f ( b ) - f ( a ) / b - a

3. The Attempt at a Solution

Since m(t) is the mass of the tumor, and we're looking for average rate of change (the slope of the secant line), we must use the MVT for derivatives.

I performed the following calculation:

m (5) - m(0) / 5 - 0 .... but got 1.8555.... not one of the answer choices.

Is my logic wrong?

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#### Mark44

Mentor
Yes, your logic is wrong. You want the average rate of change during the 5th week; i.e., between the start of week 4 and week 5. Try [m(5) - m(4)]/1. The value I get is one of those given.

#### carlodelmundo

Hi Mark44. Thank you for your quick response. Why do we find the average from the 4th week and 5th week instead of the 0th week and the 5th week? I'm having trouble grasping that concept.

I plugged it back in and get an answer of 6.546.

Thanks

#### Mark44

Mentor
What is the average rate of change, in grams per week, during the fifth week of growth?
This means from t = 4 weeks through t = 5 weeks.

What you did was the average rate of change for the first 5 weeks, not for the fifth week.

IIRC, 6.546 is what I got.

Thanks Mark

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