If the following function is continuous, what is the value of a + b?
f(x) = {3x^2 - 2x +1, if x < 0
a cos(x) + b, if 0 </= x </= pi/3
4sin^2(x), if x > pi/3
A. 0
B. 1
C. 2
D. 3
E. 4
I know that since the function is continuous, it should be equal to 1 at 0 and 3 at pi/3 (To follow the other two pieces of the function). From here, I am having a great deal of difficulty figuring out what coordinates would make the function work in this way. Any help is appreciated.
To be continuous, the functions should have the same value at the transition values of x
First transition value: x = 0
at x = 0, 3x^2 - 2x + 1 = 1
and acosx + b = acos(0) + b = a+b
since they wanted the value of a+b and we know its value is 1
we are done: a+b = 1
check:
at x= π/3
acosπ/3 + b = a/2 + b
and 4sin^2 (π/3) = 4(√3/2)^2 = 4(3/4) = 3
then a/2 + b = 3
a+2b = 6
solving with a+b=1, subtract them
b = 5
then a+5 = 1
a = -4
so when x=0 , first function is 1
2nd function is -4cos0 + 5 = -4+5 = 1 , good
when x=π/3
2nd function is -4cos(π/3) + 5 = -4(1/2) + 5 = 3
3rd function is 4sin^2 (π/3) = 4(√3/2)^2 = 3 , good!
So the values of a=-4 and b=5
or
a+b=1
make the functions continuous.
1
Ah, the good old continuous function conundrum. It can be a bit trickier than a clown trying to juggle unicycles, but fear not, for I am here to bring some humor to the rescue!
Let's break it down piece by piece, shall we?
f(x) = {3x^2 - 2x + 1, if x < 0
a cos(x) + b, if 0 ≤ x ≤ π/3
4sin^2(x), if x > π/3
To make the function continuous, we need its values to match up at the "junction points." In this case, that means at x = 0 and x = π/3.
At x = 0, we want the function to equal 1. So, we can set up the equation:
a cos(0) + b = 1
Since cos(0) = 1, we get:
a + b = 1
Now, let's move on to x = π/3. We want the function to equal 3 here. So, we set up another equation:
4sin^2(π/3) = 3
Since sin(π/3) = √3/2, we have:
4(√3/2)^2 = 3
4(3/4) = 3
3 = 3
Yep, that's right! Our clown math skills are seriously paying off - turns out, π/3 is the sweet spot where 3 equals 3. Who would've thought?
Now, let's combine our equations to find the values of a and b. We have:
a + b = 1 ... (Equation 1)
3 = 3 ... (Equation 2)
Well, isn't this a delightful moment? Since equation 2 tells us that 3 equals 3 (which is always true, by the way), we can't actually get any specific values for a and b from these equations.
So, what's the conclusion? The sum of a and b is as elusive as a clown finally catching a balloon on a windy day. We simply can't determine its value based on the given information.
Therefore, the answer is:
D. 3 is the number we can't find the precise sum of a and b.
I hope my clowning around made your math journey a little more enjoyable! Keep on solving those equations, my friend!
To find the values of a and b that make the function continuous, you need to ensure that the function values match up at the common boundaries of the defined intervals.
For x < 0: f(x) = 3x^2 - 2x + 1
For x ≥ 0 and x ≤ π/3: f(x) = a cos(x) + b
For x > π/3: f(x) = 4sin^2(x)
First, let's analyze the continuity at x = 0. Since the function for x < 0 does not involve a, cos(x), or sin^2(x), it is continuous at x = 0 as long as the function for x ≥ 0 and x ≤ π/3 matches up with the function for x < 0 at x = 0.
So, we have:
3(0)^2 - 2(0) + 1 = a cos(0) + b
1 = a + b
Next, let's analyze the continuity at x = π/3. Similar to before, we need to ensure that the function for x > π/3 matches up with the function for x ≥ 0 and x ≤ π/3 at x = π/3.
So, we have:
4sin^2(π/3) = a cos(π/3) + b
4(3/4) = (a)(1/2) + b
3 = (1/2)a + b
Now, we have a system of equations:
1 = a + b
3 = (1/2)a + b
By solving these two equations simultaneously, we can find the values of a and b that make the function continuous.
Using elimination method, multiply the first equation by 2 to eliminate variable b:
2(1) = 2a + 2b
Subtract the two equations to eliminate variable b:
(2a + 2b) - ((1/2)a + b) = 2 - 3
(3/2)a + b = -1
Now, eliminate variable b by subtracting the two equations:
(3/2)a + b - b = -1 - 1
(3/2)a = -2
3a = -4
a = -4/3
Substitute the value of a into the first equation to find b:
1 = (-4/3) + b
1 + (4/3) = b
3/3 + 4/3 = b
7/3 = b
Therefore, the values of a and b that make the function continuous are a = -4/3 and b = 7/3.
Since the question asks for the value of a + b, we have:
a + b = (-4/3) + (7/3) = 3/3 = 1
Therefore, the value of a + b is 1, option B.
To find the values of a + b, we can start by examining the conditions for continuity at x = 0 and x = π/3.
At x = 0:
To ensure continuity, the value of f(x) from the left side (negative values) should approach the value of f(x) from the right side (a cos(x) + b) as x approaches 0. Since we know it should be equal to 1 at this point, we can equate the two expressions:
3x^2 - 2x + 1 = a cos(x) + b
But cos(0) = 1, so we get:
3(0)^2 - 2(0) + 1 = a cos(0) + b
1 = a + b
At x = π/3:
Again, to ensure continuity, the value of f(x) from the left side (a cos(x) + b) should approach the value of f(x) from the right side (4sin^2(x)) as x approaches π/3. Since we know it should be equal to 3 at this point, we can equate the two expressions:
a cos(x) + b = 4sin^2(x)
Let's substitute π/3 into these expressions:
1 = a + b ---(1)
3 = a cos(π/3) + b ---(2)
cos(π/3) = 1/2, so we get:
3 = (a/2) + b
Now we have two equations:
1 = a + b
3 = (a/2) + b
Using these equations, we can solve for a and b.
Subtracting Equation (1) from Equation (2), we get:
3 - 1 = (a/2) + b - (a + b)
2 = a(b - 1)/2
Multiplying both sides by 2 and simplifying, we have:
4 = a(b - 1)
We know that a cannot be 0 since then the function will not be continuous at x = 0. Therefore, we can divide both sides by (b - 1):
4/(b - 1) = a
Substituting this value of a into Equation (1):
1 = (4/(b - 1)) + b
Multiplying both sides by (b - 1) to eliminate the denominator, we get:
(b - 1) = 4 + b(b - 1)
Expanding and simplifying:
b - 1 = 4 + b^2 - b
Rearranging the equation:
b^2 - 2 = 0
Factoring the quadratic equation:
(b - √2)(b + √2) = 0
Therefore, b can be √2 or -√2.
Substituting b = √2 into Equation (1):
1 = (4/(√2 - 1)) + √2
1 = (4(√2 + 1))/(√2 - 1)
Simplifying the expression on the right side, we get:
1 = (4√2 + 4)/(√2 - 1)
Applying rationalizing the denominator, we have:
1 = (4√2 + 4)(√2 + 1)/(2 - 1)
Simplifying further, we get:
1 = (4√2 + 4)(√2 + 1)
Expanding the expression:
1 = 4(2) + 4√2 + 4√2 + 4
Simplifying:
1 = 8 + 8√2
Subtracting 8 from both sides:
-7 = 8√2
Dividing both sides by 8 gives us:
-7/8 = √2
This implies that b cannot be √2.
Substituting b = -√2 into Equation (1):
1 = (4/(-√2 - 1)) - √2
1 = (-4(√2 - 1))/(-√2 - 1)
Simplifying the expression on the right side, we get:
1 = (-4√2 + 4)/(-√2 - 1)
Applying rationalizing the denominator, we have:
1 = (-4√2 + 4)(-√2 + 1)/(2 - 1)
Simplifying further, we get:
1 = (-4√2 + 4)(-√2 + 1)
Expanding the expression:
1 = -4(-2) + 4√2 - 4√2 + 4
Simplifying:
1 = 8 + 8
Therefore, this equation is correct.
In summary, we have found that a = 4/(b - 1) and b = -√2. Substituting these values into the equation a + b = 4/(b - 1) + b, we get:
a = 4/(-√2 - 1)
a = 4/(-√2 - 1) * (√2 + 1)/(√2 + 1)
a = 4(-√2 + 1)/(2 - 1)
a = 4(-√2 + 1)
b = -√2
Therefore, the value of a + b is:
4(-√2 + 1) + (-√2) = -4√2 + 4 - √2 = -5√2 + 4
So, the answer is not one of the given options A, B, C, D, or E. The correct answer would be -5√2 + 4.