Science:Math Exam Resources/Courses/MATH100/December 2012/Question 02 (b)

MATH100 December 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  Q6 (f)  •  Q7  •  Q8  •  Q9  •  Q10  •  Q11  •  

   Other MATH100 Exams

  •  December 2011  •  December 2010  •  December 2012  •  December 2013  •  December 2014  •  December 2015  •  December 2016  •  December 2018  •  

Question 02 (b)
Short-Answer Questions. Questions 1-4 are short-answer questions. Put your answers in the boxes provided. Simplify your answers as much as possible, and show your work. Each question is worth 3 marks, but not all questions are of equal difficulty. Find the slope of the tangent line to the curve $\displaystyle y+x\cos y=\cos x$ at the point $(0,1)$.

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
Recall the relationship between slopes of tangent lines and derivatives.

Hint 2
Implicitly differentiate with respect to x and then plug in $(0,1)$. Solve for $y'$ to get the slope since the derivative at a point is equal to the slope of the tangent line to the curve at that point.

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. Implicitly differentiating gives ${\begin{aligned}y'+\cos y+x{\tfrac {d}{dx}}(\cos y)&=-\sin x\\y'+\cos y+x(-\sin y)(y')&=-\sin x\end{aligned}}$ Plugging in $(x,y)=(0,1)$ gives ${\begin{aligned}y'+\cos(1)+0(-\sin(1))(y')&=-\sin(0)\\y'+\cos(1)&=0\\y'&=-\cos(1)\end{aligned}}$ completing the question.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Implicit differentiation, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag