Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 18

MATH100 A December 2023

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12  •  Q13  •  Q14  •  Q15  •  Q16  •  Q17  •  Q18  •  Q19  •  Q20  •  Q21  •  Q22  •  Q23  •  Q24  •  Q25  •  Q26  •  Q27(a)  •  Q27(b)  •  Q27(c)  •  Q28(a)  •  Q28(b)  •  Q29(a)  •  Q29(b)  •  Q30  •  

   Other MATH100 A Exams

  •  December 2023  •  

Question 18
At what value of ${\textstyle x}$ on the curve ${\textstyle y=x+x^{2}-x^{3}}$ does the tangent line have the largest positive slope?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
This is a maximisation problem, where you are maximizing the slope of the tangent line. What function defines the slope of the tangent line?

Hint 2
The slope of the tangent line to the curve $y=x+x^{2}-x^{3}$ at the point $(x,y)$ is the derivative of $y$ at $x$. Can you find the critical points of this function and determine whether they are maxima or minima?

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let’s start by writing an expression for the slope of the tangent line. The slope ${\textstyle m}$ of the tangent line at a point ${\textstyle x}$ is given by the derivative of the curve evaluated at $x$, namely ${\textstyle {\frac {dy}{dx}}(x)}$. We can write: $${\begin{aligned}m(x)&={\frac {dy}{dx}}(x)\\&=1+2x-3x^{2}\end{aligned}}$$ Now, to find the the maximum value of ${\textstyle m(x)}$, we need to determine the critical points, and then classify them as maximums, minimums, or saddle points. To do this, calculate the derivative of ${\textstyle m(x)}$ and set it equal to 0: $${\begin{aligned}m'(x)=2-6x&=0\\6x&=2\\x&={\frac {1}{3}}\end{aligned}}$$ The only critical point is ${\textstyle x={\frac {1}{3}}}$, and we know it is a maximum because ${\textstyle m(x)}$ (equation for the slope) is a downward facing parabola. Thus, at ${\textstyle x={\frac {1}{3}}}$ the tangent line has the largest positive slope.