Compute Δy and dy for the given values of x and dx = Δx. (Round your answers to three decimal places.)
y = 3x − x^2, x = 3, Δx = −0.6
Δy=???
at x = 3
y = 3*3 - 9 = 0
at x = 2.4
y = 3(2.4) - 2.4^2 = 1.44
delta y = 1.44 -0 = 1.44
dy/dx = 3 - 2 x
at x = 3
dy/dx = 3 - 6 = -3
if dx = -.6
dy = -3 (-.6) = 1.8
Well, well, well. It seems we have a math problem on our hands. Don't worry, I'm here to make it a little more fun.
So, we're trying to compute Δy, which represents the change in y. To do that, we need to substitute the given values of x and dx into the equation y = 3x − x^2.
Let's start with x = 3. Plugging it in, we get:
y = 3(3) − (3)^2
Simplifying that mess, we have:
y = 9 - 9
Which gives us y = 0.
Now, let's move on to Δx = -0.6. To find Δy, we need to find the change in y when x changes by -0.6.
Substituting again, we have:
y = 3(3 - 0.6) - (3 - 0.6)^2
Calculating that out, we get:
y = 3(2.4) - (2.4)^2
y = 7.2 - 5.76
Which gives us y = 1.44.
So, Δy is the difference between these values, which is:
Δy = 1.44 - 0
And finally, rounding to three decimal places, we have:
Δy = 1.440
I hope that puts a smile on your face!
To compute Δy, we need to substitute the given values of x and dx into the function y = 3x − x^2.
Given: x = 3, dx = Δx = -0.6
Thus, x + dx = 3 + (-0.6) = 2.4
Now, we can calculate the values of y for both x and x + dx:
For x = 3:
y = 3(3) - (3)^2
y = 9 - 9
y = 0
For x + dx = 2.4:
y = 3(2.4) - (2.4)^2
y = 7.2 - 5.76
y = 1.44
Finally, we can calculate Δy by subtracting the initial y from the final y:
Δy = final y - initial y
Δy = 1.44 - 0
Δy = 1.44
Therefore, Δy = 1.44 (rounded to three decimal places).
To compute Δy, we need to find the change in the value of y when x changes by Δx.
First, let's find the initial value of y when x is equal to 3.
y = 3x − x^2
y = 3(3) − (3)^2
y = 9 - 9
y = 0
Now, let's find the new value of y when x changes by Δx = -0.6.
To find the new value of x, we subtract Δx from the initial value of x.
New x = 3 + (-0.6)
New x = 2.4
Now, let's substitute the new x value into the equation to find the new y value.
y = 3x − x^2
y = 3(2.4) − (2.4)^2
y = 7.2 - 5.76
y = 1.44
Finally, to calculate Δy, we subtract the initial value of y from the new value of y.
Δy = new y - initial y
Δy = 1.44 - 0
Δy = 1.44
Therefore, Δy is equal to 1.44.