consider k(t)=(e^t)/(e^t-7) on[-7,7]
Is this function continuous on the given interval? If it is continuous, type "continuous". If not, give the t -value where the function is not continuous.
Is the t value 0
It is not continuous where the denomimator becomes zero. That happens when e^t = 7, or about t = 1.94591
To determine whether the function k(t) = (e^t)/(e^t-7) is continuous on the given interval [-7, 7], we need to analyze the function for any potential points of discontinuity.
First, let's check if there is any discontinuity at t = 0. To do this, we can evaluate the function at t = 0 by substituting the value into the function:
k(0) = (e^0)/(e^0-7) = 1/(1-7) = -1/6
Since the function is defined at t = 0 (there is no division by zero or any other undefined operations), there is no discontinuity at t = 0.
Now, let's examine if there are any potential points of discontinuity in the interval [-7, 7] by considering the denominator (e^t-7). The function is undefined when the denominator equals zero because division by zero is not defined.
(e^t-7) = 0
e^t = 7
t = ln(7)
The natural logarithm of 7 is approximately 1.95. Therefore, the only point within the given interval where the function may have a discontinuity is at t = 1.95.
However, keep in mind that this value is an approximation, so if you need a more precise answer, you may need to use numerical methods or a calculator.
In conclusion, the function k(t) = (e^t)/(e^t-7) is continuous on the given interval [-7, 7].