find K so that the following function is continuous on any interval:
f(x)=kx if 0¡Üx<3, and f(x)=2x^2 if 3¡Üx
K=
To find the value of K so that the function is continuous on any interval, we need the two parts of the function to meet at the point x = 3.
First, let's find the value of f(x) as x approaches 3 from the left side:
As 0 ≤ x < 3, the function is given by f(x) = kx.
So, when x approaches 3 from the left side, f(x) approaches f(3) = 3k.
Next, let's find the value of f(x) as x approaches 3 from the right side:
As x ≥ 3, the function is given by f(x) = 2x^2.
So, when x approaches 3 from the right side, f(x) approaches f(3) = 2(3)^2 = 2(9) = 18.
For the function to be continuous at x = 3, the left-hand limit and the right-hand limit must be equal.
Therefore, we have 3k = 18.
Dividing both sides of the equation by 3, we get:
k = 6.
So, the value of K that makes the function continuous on any interval is K = 6.
To find the value of K so that the function is continuous, we need to ensure the two parts of the function meet smoothly at the point where x=3. In other words, we need to make sure that the left and right limits at x=3 are equal.
First, let's consider the left limit as x approaches 3. We need to determine the value of f(x) when approaching 3 from the left side (0 ≤ x < 3). In this interval, f(x) = kx. So, as x approaches 3 from the left, we have:
lim(x→3-) f(x) = lim(x→3-) kx
To find the left limit, let's substitute x=3 into the equation:
lim(x→3-) f(x) = lim(x→3-) k * 3 = 3k
Now, let's consider the right limit as x approaches 3. We need to determine the value of f(x) when approaching 3 from the right side (x ≥ 3). In this interval, f(x) = 2x^2. So, as x approaches 3 from the right, we have:
lim(x→3+) f(x) = lim(x→3+) 2x^2
To find the right limit, let's substitute x=3 into the equation:
lim(x→3+) f(x) = lim(x→3+) 2 * 3^2 = 18
For the function to be continuous at x=3, the left limit and right limit must be equal. Therefore, we have the equation:
3k = 18
To find K, divide both sides by 3:
k = 18/3
Simplifying the right side:
k = 6
Hence, K = 6.