Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 28(a)

MATH100 A December 2023

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[hide] Question 28(a)
Let $$f(x)=e^{-x}.$$ The goal of this question is to find the number ${\textstyle a}$ such that the triangle consisting of the portion of the first quadrant that lies below the tangent line to ${\textstyle f(x)}$ at ${\textstyle x=a}$ has the largest possible area. Part A: Draw a large sketch of the function and the region in question, making sure to label the value ${\textstyle x=a}$ .

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
Is ${\textstyle f(x)}$ increasing or decreasing with ${\textstyle x}$? Find also the inflection points to determine the concavity of the graph.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution

Since $e^{x}>0$ for all $x$, it follows that the derivative, given by $f'(x)=-e^{-x}$, is negative for all $x$, and the second derivative, which is equal to $f$ itself is positive. Thus the graph of the function is decreasing, concave up and has no critical or inflection points. Taking the limit as $x\rightarrow \infty$ reveals that it has a horizontal asymptote $y=0$. The plot for the function and the tangent line is given below. The area is that enclosed by the the triangle whose vertices are the origin and the two intersections of the tangent line with the x- and y-axes. Function and tangent line.