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 14.

t= ____

f(t) = 1/t

f(1) = 1

( 1/t - 1)/(t-1) = 14 , where t ≠ 1
1/t - 1 = 14t - 14
times t
1 - t = 14t^2 - 14t
14t^2 - 13t - 1 = 0
(14t + 1)(t - 1) = 0

t = -1/14

To find the value of t such that the average rate of change of f(t) from 1 to t equals 14, we need to use the formula for average rate of change:

Average rate of change = (f(t) - f(1)) / (t - 1)

In this case, f(t) = 1/t, so we have:

Average rate of change = (1/t - 1) / (t - 1)

We want to find a value of t such that the average rate of change is equal to 14, so we can set up the equation:

14 = (1/t - 1) / (t - 1)

To solve this equation, we can first cross-multiply:

14(t - 1) = 1 - t

14t - 14 = 1 - t

Bringing all the terms with t to one side:

14t + t = 1 + 14

15t = 15

Dividing both sides by 15:

t = 1

Therefore, the value of t that satisfies the equation is t = 1.

To find a value of t such that the average rate of change of f(t) from 1 to t equals 14, we need to compute the average rate of change of f(t) over the interval [1, t] and set it equal to 14.

The average rate of change of a function f(t) over an interval [a, b] is given by the formula:

Average rate of change = (f(b) - f(a))/(b - a)

In this case, we need to find t such that the average rate of change of f(t) over the interval [1, t] equals 14. So we have:

14 = (f(t) - f(1))/(t - 1)

Substituting the function f(t) = 1/t into the equation, we get:

14 = (1/t - 1)/(t - 1)

To simplify the equation, we can multiply both sides by (t - 1) to eliminate the denominator:

14(t - 1) = 1 - t

Expanding the left side of the equation:

14t - 14 = 1 - t

Combining like terms:

15t = 15

Dividing both sides by 15:

t = 1

Therefore, the value of t that satisfies the equation and makes the average rate of change of f(t) from 1 to t equal to 14 is t = 1.