Compute Δy and dy for the given values of x and
dx = Δx.
y = x^2 − 7x, x = 6, Δx = 0.5
dy = (2x-7) dx
so,
Δx ≈ (2x-7) Δx
Well, well, well! Looks like we have some calculations to do here. Let's get to it!
To find Δy, we need to calculate the change in y when x changes by Δx. So here's what we do:
First, we substitute x = 6 into the equation for y to get the initial value of y:
y = (6)^2 - 7(6)
Now let's calculate that:
y = 36 - 42
y = -6
Now, we increase x by Δx = 0.5. This means x becomes 6 + 0.5 = 6.5. And we need to find the new value of y for that:
y_new = (6.5)^2 - 7(6.5)
y_new = 42.25 - 45.5
y_new = -3.25
Finally, we can calculate Δy by subtracting the initial y from the new y:
Δy = y_new - y
Δy = -3.25 - (-6)
Δy = -3.25 + 6
Δy = 2.75
So, Δy = 2.75. And that's the change in y for the given values of x and Δx.
As for dy, it represents the derivative of y with respect to x. It tells us the rate of change of y with respect to x, for small changes in x. To calculate dy, we can use the power rule of differentiation:
dy/dx = 2x - 7
Substituting x = 6 into this equation gives us:
dy/dx = 2(6) - 7
dy/dx = 12 - 7
dy/dx = 5
Therefore, dy = 5. And that's the derivative of y with respect to x for the given values.
Hope that clarifies things for you! I'm here all day if you need more humor-infused assistance.
To compute Δy, we need to first substitute the given values of x and dx into the equation and calculate y:
Let's calculate y when x = 6:
y = x^2 − 7x
y = 6^2 − 7(6)
y = 36 - 42
y = -6
Now, let's calculate y when x = 6 + Δx = 6 + 0.5 = 6.5:
y = (6.5)^2 − 7(6.5)
y = 42.25 - 45.5
y ≈ -3.25
Δy is the difference in y values, which is given by:
Δy = y - y_initial
Δy = (-3.25) - (-6)
Δy = -3.25 + 6
Δy = 2.75
Lastly, dy represents the change in y values corresponding to the change in x values, which is given by:
dy = Δy / Δx
dy = 2.75 / 0.5
dy = 5.5
Therefore, Δy is 2.75 and dy is 5.5 for the given values of x = 6, dx = Δx = 0.5.
To compute Δy and dy for the given values of x, dx, and y, we can use the formulas:
Δy = y(x + Δx) - y(x)
dy = Δy / Δx
First, let's substitute the given values into the formula for y:
y = x^2 − 7x
Substituting x = 6 into the formula, we get:
y = 6^2 - 7(6)
y = 36 - 42
y = -6
Next, we can substitute x = 6 and Δx = 0.5 into the formula for Δy:
Δy = y(x + Δx) - y(x)
Δy = y(6 + 0.5) - y(6)
Δy = y(6.5) - y(6)
Δy = (6.5)^2 - 7(6.5) - (-6)
Simplifying this, we get:
Δy = 42.25 - 45.5 - (-6)
Δy = 42.25 - 45.5 + 6
Δy = 3.75
Finally, to find dy, we can use the formula:
dy = Δy / Δx
Substituting the values we found:
dy = 3.75 / 0.5
dy = 7.5
Therefore, Δy = 3.75 and dy = 7.5 for the given values of x = 6, dx = Δx = 0.5, and y = x^2 − 7x.