MATH110 April 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 • Q6 • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q8 (e) • Q8 (f) • Q9 •
Question 01 (a)
Determine whether this statement is true. If it is, explain why; if not, give a counterexample.
A linear approximation L to a function ƒ satisfies L(x) = f(x) for exactly one value of x.
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!
when the function and the linear approximation intersect. Can a linear approximation and a function intersect in more than one place?
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.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
The statement is false, so a linear approximation to a function may satisfy for more than one value of . To prove this, we need an example of such a function. In particular, we would like to find a function with a tangent line (the linear approxmiation) that intersects the function at another point.
One example would be the function , with the linear approximation at the point, say, . Drawing that function and linear approximation looks like this:
So we see that both at and at . So the original statement is false.
If you want to make it even simpler, choose ƒ to be linear, e.g. f(x) = 17x+42. Then the linear approximation L coincides with ƒ at all points.
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Linear approximation, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag