Find the values for a and b such that f(x) is continuous on the interval 0 < or = to x < or = to 13.
{ 2x^2, 0 < or = to x < or = to 3
f(x) = { ax + 3, 3 < or = to x < or = to 7
{ b, 7 < or = to x < or = to 13
Can someone explain? Am I suppose to find the x, a, and b that would make all three sets equal to the same number?
Correction:
f(x) = { 2x^2, 0 < or = to x < 3
.......{ ax + 3, 3 < or = to x < 7
.......{ b, 7 < or = to x < or = to 13
the ....... is to align them, sorry.
If it is continous, then
2x^2= ax+3 at x=3 put x=3, and solve for a.
And, b= ax+3 at x=7, solve for b
Thank you, bobpursley! I didn't know that that is what they meant by continuous.
So a=5 and b=38?
To find the values of a and b such that f(x) is continuous on the interval 0 ≤ x ≤ 13, we need to make sure that the function values match at the points where the intervals change.
First, let's examine the conditions for continuity at the boundaries of the intervals:
1. At x = 3: We need the function value f(x) to be the same from both directions. So we set 2x^2 = ax + 3 when x = 3.
2. At x = 7: Again, we need the function value f(x) to be continuous from both directions. Here, we set ax + 3 = b when x = 7.
Now, let's solve these equations to find the values of a and b.
1. At x = 3:
2(3)^2 = a(3) + 3
18 = 3a + 3
3a = 15
a = 5
So, we have found the value of a to be 5.
2. At x = 7:
a(7) + 3 = b
35 + 3 = b
b = 38
So, we have found the value of b to be 38.
Therefore, the values of a and b that make f(x) continuous on the interval 0 ≤ x ≤ 13 are a = 5 and b = 38.