Question

Let f(t) = 1/t. Find a value of t such that the average rate of change of f(t) from 1 to t equals -1/13.

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Fred

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85 Answers

Solution:

1. The average rate of change of a functionf(t) from t = a to t = b is given by:

\frac{f(b) - f(a)}{b - a}

2. Given:

* Function:f(t) = \frac{1}{t}

* Interval:[1, t]

* Average rate of change:-\frac{1}{13}

3. Substitute the given values into the formula:

\frac{f(t) - f(1)}{t - 1} = -\frac{1}{13}

4. Evaluatef(1) :

f(1) = \frac{1}{1} = 1

5. Substitutef(1) and f(t) :

\frac{\frac{1}{t} - 1}{t - 1} = -\frac{1}{13}

6. Simplify the numerator:

\frac{1 - t}{t(t - 1)} = -\frac{1}{13}

7. Multiply both sides byt(t - 1) :

1 - t = -\frac{t(t - 1)}{13}

8. Distribute and simplify:

1 - t = -\frac{t^2 - t}{13}

13(1 - t) = -t^2 + t

13 - 13t = -t^2 + t

9. Rearrange into a standard quadratic equation:

t^2 - 14t + 13 = 0

10. Factor the quadratic equation:

(t - 13)(t - 1) = 0

11. Solve fort :

t = 13 \quad \text{or} \quad t = 1

12. Since we are looking for the interval from1 to t and t = 1 does not change the interval, we discard t = 1 .

Therefore, the solution ist = 13 .

1. The average rate of change of a function

2. Given:

* Function:

* Interval:

* Average rate of change:

3. Substitute the given values into the formula:

4. Evaluate

5. Substitute

6. Simplify the numerator:

7. Multiply both sides by

8. Distribute and simplify:

9. Rearrange into a standard quadratic equation:

10. Factor the quadratic equation:

11. Solve for

12. Since we are looking for the interval from

Therefore, the solution is

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