A patient's pulse measures 110 bpm, 70 bpm, then 60 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression
(x − 110)2 + (x − 70)2 + (x − 60)2.
What value (in bpm) minimizes it?
C=(x − 110)2 + (x − 70)2 + (x − 60)2.
dC/x=2(x-110)+2(x-70) + 2(x-60)=0
6x=220+140+120
x=480/6 = ...
Well, you've come to the right bot for a clownish answer to this mathematical question! Now, let me put on my oversized spectacles and think about this for a second.
To minimize this expression, we can start by expanding and simplifying it:
(x − 110)² + (x − 70)² + (x − 60)²
= x² - 220x + 12100 + x² - 140x + 4900 + x² - 120x + 3600
= 3x² - 480x + 20600
To find the value that minimizes this expression, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula: x = -b / (2a)
In this case, a = 3 and b = -480. Plugging in these values, we get:
x = -(-480) / (2 * 3)
x = 480 / 6
x = 80
So, the value that minimizes the expression is 80 bpm. Ta-da! Now you can go back to your doctor with this dazzling mathematical result.
To find the value that minimizes the expression (x − 110)2 + (x − 70)2 + (x − 60)2, we can take the derivative of the expression with respect to x and set it equal to zero.
Taking the derivative:
d/dx [(x − 110)2 + (x − 70)2 + (x − 60)2] = 2(x − 110) + 2(x − 70) + 2(x − 60)
Setting it equal to zero:
2(x − 110) + 2(x − 70) + 2(x − 60) = 0
Simplifying:
2x − 220 + 2x − 140 + 2x − 120 = 0
6x − 480 = 0
Solving for x:
6x = 480
x = 480/6
x = 80
Therefore, the value that minimizes the expression is 80 bpm.
To find the value that minimizes the expression, we need to take the derivative of the expression with respect to x and set it equal to zero. Let's go through the steps:
1. Start with the expression: (x − 110)² + (x − 70)² + (x − 60)².
2. Expand the squares:
(x² - 220x + 12100) + (x² - 140x + 4900) + (x² - 120x + 3600).
3. Combine like terms:
3x² - 480x + 20600.
4. Take the derivative of this expression with respect to x:
d/dx(3x² - 480x + 20600) = 6x - 480.
5. Set the derivative equal to zero and solve for x:
6x - 480 = 0,
6x = 480,
x = 80.
Therefore, the value that minimizes the expression is x = 80 bpm.