At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.

3x2y - π cos y = 4π, slope at (1, π)

Question

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.

3x2y - π cos y = 4π, slope at (1, π)

BlackTom AI · Accepted Answer

The problem asks for the slope of the tangent to the curve defined by 3x^2 y − π cos y = 4π at the point (1, π).

First, I’ll differentiate the given implicitly with respect to x. For the term 3x^2 y, use the product rule: d/dx(3x^2 y) = 6x y + 3x^2 y'. For the term −π cos y, apply the chain rule: d/dx(−π cos y) = −π(−sin y) y' = π sin y · y'. The constant 4π differentiates to 0. Putting these together gives:

6x y + 3x^2 y' + π sin y · y' = 0.

Now factor y' on the left:
(3x^2 + π sin y) y' = −6x y.

Therefore, y' = slope of the tangent = −6x y / (3x^2 + π sin y).

Next, substitute the given point (x, y) = (1, π). Since sin(π) = 0, the denominator becomes 3(1)^2 + π·0 = 3. The numerator becomes −6·1·π = −6π. Thus: y' = (−6π)/3 = −2π.

Now examine the answer options:
- Option 1: 0. This would require either the numerator to be 0 or the denominator to be infinite, neither of which happens with the computed expression at (1, π). This option ignores the actual interplay of x, y, and their derivatives and is therefore incorrect.
- Option 2: −π/2. This value does not match the result of the calculation above. A common pitfall would be misapplying the quotient or product rule or plugging in the point incorrectly, but the correct differentiation steps lead away from this value.
- Option 3: π. This is positive, whereas the slope we computed is negative. The sign discrepancy already signals a mismatch with the actual slope as derived from implicit differentiation.
- Option 4: −2π. This exactly matches the slope obtained from the differentiation and substitution at (1, π). If one were to miscalculate the denominator as something other than 3 (e.g., forgetting that sin π = 0), one might land elsewhere, but with sin y evaluated at y = π, the denominator is 3, giving −2π.

In summary, the calculation shows that the tangent slope at (1, π) is −2π. The other options fail for the following reasons: 0 ignores the nonzero numerator; −π/2 miscomputes either the numerator or the denominator; π has the wrong sign; and only −2π matches the properly derived result.

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. 3x2y - π cos y = 4π, slope at (1, π) 单项选择题

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