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 : d x = 2.02 − 2 = 0.02 .
Approximate the change in y : d 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 about finding the approximate change in a function using derivatives.
Using Differentials To find the approximate increase in y , we can use the differential d y , which is given by d y = d x d y d x , where d x d y is the derivative of y with respect to x , and d x is the change in x .
Finding the Derivative First, we need to find the derivative of y = x 3 with respect to x . Using the power rule, we have d x d y = 3 x 2
Evaluating the Derivative Next, we evaluate the derivative at x = 2 :
d x d y ∣ x = 2 = 3 ( 2 ) 2 = 3 ( 4 ) = 12
Finding the Change in x Now, we find the change in x , which is d x = 2.02 − 2 = 0.02 .
Calculating the Approximate Change in y Finally, we can find the approximate change in y :
d y = d x d y d x = 12 × 0.02 = 0.24
Final Answer Therefore, the approximate increase in y when x increases from 2 to 2.02 is 0.24.
Examples
Imagine you are designing a bridge and need to calculate how much the length of a steel beam will change when the temperature increases slightly. Using derivatives, you can approximate this change. If the length L of the beam is a function of temperature T , then the change in length Δ L can be approximated by d T d L Δ T , where d T d L is the rate of change of length with respect to temperature, and Δ T is the change in temperature. This allows engineers to make accurate predictions and ensure the structural integrity of the bridge.
The approximate increase in y when x increases from 2 to 2.02 for the function y = x^3 is calculated using derivatives, resulting in an increase of 0.24. Therefore, the correct choice is option A. 0.24.
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