Methods2024 NHT Exam 1 · Q9Tech-free8 marks

Derivative, transformations and area between a curve and its tangent

Let $f: (0, \infty) \to \R, f(x) = (x - 1)e^{-x}$ .

Find $f'(x)$ .

1 mark

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State a sequence of transformations that will map $f(x)$ onto $f'(x)$ .

2 marks

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Show that $\frac{d}{dx}\left(-xe^{-x}\right) = f(x)$ .

1 mark

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Let $y = g(x)$ be the tangent to $y = f(x)$ at the point $\left(3, f(3)\right)$ .

Using $\frac{d}{dx}\left(-xe^{-x}\right) = f(x)$ , or otherwise, determine the area of the region bounded by the lines $x = 1$ , $y = g(x)$ and the graph of $y = f(x)$ .

Graph of y = f(x) for x > 0, rising steeply through (1, 0) to a maximum and then decreasing towards the x-axis, with the tangent line y = g(x) touching the curve at the marked point (3, f(3)). The vertical line x = 1 is drawn, and the region between x = 1, the tangent line and the curve is shaded.

4 marks

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