Consider the function :[1,∞)→Rgivenbytheformulay=x3+2.

(a) Show that though the function is increasing at an increasing rate, the change in x required to double the value of the function is not constant.

(b) Make a table of values for constant changes in x, and investigate whether or not the property ∆y ∝ y is satisfied.

heck, the answer to the 2nd part provides the answer to the first part:

x y ∆y
1 3
2 10 7
3 29 19
4 66 37
5 127 61
6 218 91

as you can see, x has to increase less and less for y to double.

Thank you very much Steve..

(a) To show that the function is increasing at an increasing rate, we need to demonstrate that its derivative is positive and increasing. The derivative of the function y = x^3 + 2 can be found by applying the power rule of differentiation. Thus, dy/dx = 3x^2.

Now, we need to analyze the change in x required to double the value of the function. Let's assume y1 = 2y0, where y0 represents the initial value of the function and y1 represents double the value of the function. Plugging in y0 and y1 into the function, we have:

y0 = x0^3 + 2
y1 = x1^3 + 2

Dividing y1 by y0:

2 = (x1^3 + 2) / (x0^3 + 2)

Simplifying, we have:

2(x0^3 + 2) = x1^3 + 2

Expanding and rearranging the equation:

2x0^3 + 4 = x1^3

Now, let's analyze the change in x required to double the value of the function. Taking the cube root of both sides, we get:

x1 = (2x0^3 + 4)^(1/3)

Since x1 is a function of x0, the change in x required to double the value of the function is not constant, as it depends on the initial value x0.

(b) To investigate whether or not the property ∆y ∝ y is satisfied, we need to examine whether the change in y (∆y) is proportional to y for constant changes in x.

Let's create a table of values for constant changes in x and observe the changes in y. Suppose we have three values for x: x1, x2, and x3, with x2 = x1 + Δx and x3 = x1 + 2Δx (where Δx represents a constant change in x). Plugging these values into the function, we get:

y1 = x1^3 + 2
y2 = x2^3 + 2 = (x1 + Δx)^3 + 2
y3 = x3^3 + 2 = (x1 + 2Δx)^3 + 2

Now, let's calculate the changes in y:

∆y1 = y2 - y1 = [(x1 + Δx)^3 + 2] - [x1^3 + 2]
∆y2 = y3 - y1 = [(x1 + 2Δx)^3 + 2] - [x1^3 + 2]

To determine whether ∆y ∝ y, we need to check if the change in y (∆y) is proportional to y. This means that for a given constant Δx, if we divide ∆y by y, the result should be constant. Let's calculate this ratio for the values we obtained:

∆y1/y1 = [(x1 + Δx)^3 + 2 - (x1^3 + 2)] / (x1^3 + 2)
∆y2/y1 = [(x1 + 2Δx)^3 + 2 - (x1^3 + 2)] / (x1^3 + 2)

By simplifying these expressions and checking if the ratios are constant for different values of x1 and Δx, we can determine if ∆y ∝ y is satisfied.