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.