Science:Math Exam Resources/Courses/MATH200/April 2012/Question 02 (a)

MATH200 April 2012

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

   Other MATH200 Exams

  •  April 2012  •  December 2011  •  

Question 02 (a)
Let $\displaystyle {}z=f(x,y)=\ln(4x^{2}+y^{2})$ Use a linear approximation of the function z=f(x,y) at (0,1) to estimate f(0.1,1.2).

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!

Hint
The linear approximation to a multivariable function ƒ(x,y) at a point (x,y) = (a,b) is given by L(x,y) where $\displaystyle L(x,y)=f(a,b)+f_{x}(a,b)(x-a)+f_{y}(a,b)(y-b)$ where the subscripts denote partial differentiation with respect to the given variable.

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.

Solution

The linear approximation to a multivariable function ƒ(x,y) at a point (x,y) = (a,b) is given by L(x,y) where $\displaystyle L(x,y)=f(a,b)+f_{x}(a,b)(x-a)+f_{y}(a,b)(y-b)$ where the subscripts denote partial differentiation with respect to the given variable. In this case the partial derivatives of $f(x,y)=\ln(4x^{2}+y^{2})$ are $f_{x}(x,y)={\frac {8x}{4x^{2}+y^{2}}},\quad f_{y}(x,y)={\frac {2y}{4x^{2}+y^{2}}}.$ and we wish to approximate ƒ near (x,y) = (0,1). Hence the linear approximation we seek is: ${\begin{aligned}\displaystyle L(x,y)&=\ln(1)+0\cdot (x-0)+2(y-1)\\&=2y-2\end{aligned}}$ Using the linear approximation, we can estimate the value of ƒ(0.1,1.2) as ${\color {blue}f(0.1,1.2)\approx L(0.1,1.2)=0.4}.$

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Linear approximation, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag