Given $y =x^3$, find the approximate increase in y when x increases from 2 to 2.02
(a) 0.24
(b) 0.12
(c) 0.48
(d) 0.04
Answer (2)
Find the derivative of y = x 3 , which is d x d y = 3 x 2 .
Evaluate the derivative at x = 2 : d x d y ∣ x = 2 = 12 .
Calculate the change in x : Δ x = 2.02 − 2 = 0.02 .
Approximate the increase in y : Δ y ≈ 12 × 0.02 = 0.24 .
Explanation
Problem Analysis We are given the function y = x 3 and asked to find the approximate increase in y when x increases from 2 to 2.02. This is a problem that can be solved using the concept of derivatives. The derivative of a function gives us the rate of change of the function with respect to its variable. In this case, we want to find how much y changes when x changes by a small amount.
Finding the Derivative First, we need to find the derivative of y with respect to x . Given y = x 3 , we can use the power rule to find the derivative: d x d y = 3 x 2
Evaluating the Derivative Next, we need to evaluate the derivative at x = 2 . This will give us the rate of change of y at that specific point: d x d y ∣ x = 2 = 3 ( 2 ) 2 = 3 ( 4 ) = 12
Approximating the Change in y Now, we can approximate the change in y ( Δ y ) using the formula: Δ y ≈ d x d y ∣ x = 2 ⋅ Δ x where Δ x is the change in x , which is 2.02 − 2 = 0.02 .
Calculating the Approximate Increase Plugging in the values, we get: Δ y ≈ 12 ⋅ 0.02 = 0.24 So, the approximate increase in y is 0.24.
Examples
In physics, if you have a formula for the distance an object travels as a function of time, you can use derivatives to find the object's velocity at a particular moment. Similarly, in economics, if you have a cost function, you can use derivatives to find the marginal cost, which is the approximate cost of producing one more unit. This type of calculation is also used in engineering to optimize designs and predict how systems will respond to small changes.
The approximate increase in y when x increases from 2 to 2.02 for the function y = x 3 is 0.24. The chosen option is (a) 0.24.
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