Use the information to evaluate and compare Δy and dy. (Round your answers to three decimal places.)
y = x4
x = 1
Δx = dx = 0.1
I'm totally lost
To evaluate and compare Δy and dy, we need to use the given information:
y = x^4
x = 1
Δx = dx = 0.1
First, let's find the value of y when x = 1:
y = (1)^4 = 1
Now, let's find Δy by plugging in x + Δx into the equation:
Δy = (x + Δx)^4
Δy = (1 + 0.1)^4
Δy = (1.1)^4
Δy ≈ 1.464
Next, let's find dy by plugging in dx into the equation:
dy = (dx)^4
dy = (0.1)^4
dy = 0.0001
Finally, we can compare Δy and dy:
Δy ≈ 1.464
dy = 0.0001
From this comparison, we can see that Δy is significantly larger than dy.
To evaluate and compare Δy and dy, let's start by understanding what each of them represents.
Δy represents the change in the value of y when x changes by a certain amount, which in this case is Δx or dx.
dy, on the other hand, represents the instantaneous change in the value of y with respect to a very small change in x. It can be thought of as the slope of the tangent line to the curve y = x^4 at a specific point.
To compute Δy, we can use the formula:
Δy = y(x + Δx) - y(x)
Here, y(x) represents the value of y when x is 1, and Δx or dx is given as 0.1. Let's calculate it:
y(x + Δx) = (x + Δx)^4 = (1 + 0.1)^4 = 1.1041
y(x) = x^4 = 1^4 = 1
Δy = y(x + Δx) - y(x) = 1.1041 - 1 = 0.1041
So, Δy is 0.1041.
To compute dy, we need to take the derivative of y (dy/dx) and evaluate it at x = 1. The derivative of y = x^4 is:
dy/dx = 4x^3
Substituting x = 1 into the derivative:
dy = 4(1^3) = 4
So, dy is 4.
To summarize:
Δy = 0.1041
dy = 4
Therefore, Δy and dy are different values that represent different measures of change in y with respect to x.
Δy = f(1.1)-f(1) = 0.4641
Δy ≈ y' Δx = 4x^3*0.1 = 0.41
you are low because the graph is concave up at x=1. The y' approximation follows the tangent line at x=1, rather than going exactly along the curve.
y=x^4
dy=4x^3 dx=4(1)(.1)=.4
deltay= (x+dx)^4-x^4
= x^4+4dx*x^3+6dx^2*x^2+4dx^3*x+ dx^4 -x^4
check that
= 4(.1)1+6(.01)1 + 4(.001)1+(.0001)=
=.4641 check