Find the value of the constant c that makes the following function continuous on (−∞,∞).
f(t)= ct+8 if t∈(−∞,6]
ct^2−8 if t∈(6,∞)
If t=6
The two function are said to be equal for continuity
36c-8=6c+8
I already did like that but it is wrong
Oh god it was correct thanks
How is it correct, what would the value of C be
Well, to find the value of the constant c that makes the function continuous, we need to ensure that the limit as t approaches 6 from the left is equal to the limit as t approaches 6 from the right.
So, let's find those limits:
As t approaches 6 from the left, we have:
lim (t→6-) f(t) = lim (t→6-) (ct + 8) = 6c + 8
As t approaches 6 from the right, we have:
lim (t→6+) f(t) = lim (t→6+) (ct^2 - 8) = c(6^2) - 8 = 36c - 8
To make the function continuous, both limits must be equal. Therefore, we get the equation:
6c + 8 = 36c - 8
Solving this equation will give us the value of c. So, let's do that:
First, let's isolate c on one side of the equation:
6c - 36c = -8 - 8
-30c = -16
Now, let's solve for c:
c = (-16) / (-30)
c = 8/15
So, the value of the constant c that makes the function continuous on (-∞,∞) is 8/15.
But hey, don't worry, even if the function wasn't continuous, I'd still be here to entertain you with jokes!
To find the value of the constant c that makes the function continuous on (-∞, ∞), we need to ensure that the function is continuous at the point where the two pieces of the function meet, which is when t = 6.
For a function to be continuous at a point, the values of the left-hand limit and the right-hand limit at that point should be equal to the value of the function at that point.
So let's calculate the left-hand limit, the right-hand limit, and the value of the function at t = 6.
Left-hand limit (as t approaches 6):
lim (t→6-) f(t) = lim (t→6-) (ct + 8) = 6c + 8
Right-hand limit (as t approaches 6):
lim (t→6+) f(t) = lim (t→6+) (ct^2 - 8) = 36c - 8
Value of the function at t = 6:
f(6) = 6c^2 - 8
Since the function should be continuous at t = 6, the left-hand limit, the right-hand limit, and the value of the function at t = 6 should all be equal. Therefore, we can set up the following equation:
6c + 8 = 36c - 8 = 6c^2 - 8
Solving this equation will give us the value of c that makes the function continuous on (-∞, ∞).