Find the differential dy and evaluate dy for the given values of x and dx.
y=cosx
x=pi/3
dx=0.05
2)
y=e^x/10, x=0, dx=0.1
This is a different format than my teacher used in class so I'm confused. Help please!
To find the differential dy, we first find the derivative of the given function with respect to x.
For the first example,
Given: y = cos(x)
To find dy, we take the derivative of y with respect to x and multiply it by dx:
dy = (d/dx)(cos(x)) * dx
The derivative of cos(x) is -sin(x), so:
dy = -sin(x) * dx
Now, let's evaluate dy for the given values of x and dx:
x = π/3
dx = 0.05
Substituting these values into the equation for dy, we have:
dy = -sin(π/3) * 0.05
To evaluate this, solve for sin(π/3) using the unit circle or a calculator:
sin(π/3) = √3/2
Now, substitute this value back into the equation:
dy = -(√3/2) * 0.05
To simplify, multiply:
dy = -√3/2 * 0.05
So, dy ≈ -0.025√3
For the second example,
Given: y = e^x/10
To find dy, we take the derivative of y with respect to x and multiply it by dx:
dy = (d/dx)(e^x/10) * dx
Using the chain rule to differentiate e^x, we have:
dy = (1/10)(e^x) * dx
Now, let's evaluate dy for the given values of x and dx:
x = 0
dx = 0.1
Substituting these values into the equation for dy, we have:
dy = (1/10)(e^0) * 0.1
Remember that e^0 is equal to 1:
dy = (1/10)(1) * 0.1
To simplify, multiply:
dy = 1/100
So, dy = 0.01
I hope this helps clarify how to find the differential dy and evaluate it for a given function and values of x and dx.