Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (i)

MATH104 December 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q1 (o)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  

   Other MATH104 Exams

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

Question 01 (i)
Let $\displaystyle {}p(q)=e^{3q}.$ Consider its 20th order Taylor polynomial at q=1; $T_{20}(q)=a_{0}+a_{1}(q-1)+a_{2}(q-1)^{2}+\dots +a_{20}(q-1)^{20}$ What is $a_{20}$?

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 (or review) the definition of the 20th order Taylor polynomial of a function $p(q)$ at a point $q=a$ from your text (don't let the change of notation scare you! In this case, f is p and x is q).

Hint 2

Recall that the second order Taylor polynomial of a function $p(q)$ at a point $q=a$ is given by $T_{20}(x)$ where, $\displaystyle T_{20}(q)=p(a)+p'(a)(q-a)+{\frac {1}{2}}p''(a)(q-a)^{2}+..+{\frac {1}{20!}}p^{(20)}(a)(q-a)^{20}$ In our case, $a=1$ and $p(q)=e^{3q}$. However even knowing this, it might not be clear how to proceed - Look at what $a_{20}$ is in the equation above and try to figure out a pattern to help you compute this value.

Hint 3
The previous hint tells us that $\displaystyle a_{20}={\frac {p^{(20)}(1)}{20!}}$ So we need to compute the 20th derivative of $p(q)$ at the point $q=1$. Try computing the first few derivatives and seeing if you can figure out the pattern.

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. Proceeding as in the hints, we compute the 20th derivative to be ${\begin{aligned}p'(q)&=3e^{3q}\\p''(q)&=3^{2}e^{3q}\\p'''(q)&=3^{3}e^{3q}\\&\vdots \\p^{(20)}(q)&=3^{20}e^{3q}\\\end{aligned}}$ Now using the second hint, we see that $\displaystyle a_{20}={\frac {p^{(20)}(1)}{20!}}={\frac {3^{20}e^{3}}{20!}}$ which completes the problem.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Taylor series, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag