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جستجوهای پرتکرار

میتونی لایو بذاری!

The diagram shows the curve $y = {x^2}{e^{ - x}}$.

a) Show that the area of the shaded region bounded by the curve, the x-axis and the line $x = 3$ is equal to $2 - \frac{{17}}{{{e^3}}}$.

b) Find the x-coordinate of the maximum point $M$ on the curve.

c) Find the x-coordinate of the point $P$ at which the tangent to the curve passes through the origin.

پاسخ تشریحی :
نمایش پاسخ

a) Attempt integration by parts and reach $ \pm {x^2}{e^{ - x}} \pm \int {2x{e^{ - x}}dx} $

Obtain $ \pm {x^2}{e^{ - x}} + \int {2x{e^{ - x}}dx} $, or equivalent

Integrate and obtain $ - {x^2}{e^{ - x}} - 2x{e^{ - x}} - 2{e^{ - x}}$, or equivalent

Use limits $x = 0$ and $x = 3$, having integrated by parts twice

Obtain the given answer correctly

b) Use correct product or quotient rule

Obtain correct derivative in any form

Equate derivative to zero and solve for non-zero $x$

Obtain $x = 2$ with no errors send

c) Carry out a complete method for finding the x-coordinate of $P$

Obtain answer $x = 1$

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