Question at position 8 If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x + 2.3x2 - 3x.3x2 - 3x - 2.6.

Question

Question at position 8  If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x + 2.3x2 - 3x.3x2 - 3x - 2.6.

BlackTom AI · Accepted Answer

Let me restate the problem to anchor our work: We are given y' = 6x - 3 and the initial condition y(2) = 4. We need to find the function y in terms of x and then compare against the provided answer choices.

Option 1: 6x^2 - 3x + 2. This would come from integrating y' to obtain y = 3x^2 - 3x plus a constant, not 6x^2. The coefficient of x^2 here is 6 instead of 3, so this option is inconsistent with the derivative given. Moreover, evaluating at x = 2 would yield y(2) = 6*(4) - 3*(2) + 2 = 24 - 6 + 2 = 20, which does not match y(2) = 4. This makes it incorrect.

Option 2: 3x^2 - 3x + 2. This has the correct x^2 and x terms from integrating y' (since ∫(6x - 3) dx = 3x^2 - 3x). However, the constant term here is +2, whereas the initial condition y(2) = 4 requires a different constant. If you plug x = 2, you get y(2) = 3*(4) - 3*(2) + 2 = 12 - 6 + 2 = 8, which does not equal 4. Therefore, this is not the right constant.

Option 3: 3x^2 - 3x. This expression lacks the constant term entirely. While the antiderivative form 3x^2 - 3x is correct up to a constant, applying the initial condition y(2) = 4 gives y(2) = 3*(4) - 3*(2) = 12 - 6 = 6, which is not 4. Thus, this cannot be the correct full solution.

Option 4: 3x^2 - 3x - 2. This option matches the antiderivative form 3x^2 - 3x plus a constant C, where C must be determined from the initial condition. Substituting y(2) = 4 gives 3*(4) - 3*(2) + C = 12 - 6 + C = 6 + C = 4, so C = -2. Therefore y = 3x^2 - 3x - 2 satisfies both the derivative and the initial condition, making it the consistent solution.

Option 5: 6. This is a completely different constant and does not even resemble the proper antiderivative form of 6x - 3. It fails to account for the x-dependence in y, and evaluating at any x would not align with the required behavior from y'. Hence, it is incorrect.

In summary, each option was checked against the fundamental steps: integrate y' to get the general antiderivative, then apply y(2) = 4 to determine the constant, and finally compare the resulting expression to the provided choices. The comparison shows that only the fourth option aligns with both the derivative and the initial condition. The remaining options fail due to incorrect coefficients, missing or incorrect constants, or mismatch with the initial condition.

Question at position 8 If y' = 6x - 3 and y(2) = 4, then y =6x2 - 3x + 2.3x2 - 3x + 2.3x2 - 3x.3x2 - 3x - 2.6.

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