# Exam-Style Question on Calculus

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 569. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.

Consider the function $$f$$ defined by $$f(x) = 25e^{x-5}$$ for $$x \in \mathbb{R}^+$$.

(a) Find the coordinates of the points where the graph of $$f$$ intersects the line $$y=x$$.

The line $$L$$ has a gradient of $$-1$$ and is a normal to the graph of $$f$$ at the point $$R$$.

(b) Find the exact coordinates of $$R$$.

(c) Show that the equation of the line $$L$$ is $$y=-x+6- \ln{25}$$.

(d) Find the area of the region enclosed by the graph of $$f$$ and its inverse.

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