Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 05 (c)
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 • Q7 • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q10 (a) • Q10 (b) • Q10 (c) • Q11 (a) • Q11 (b) • Q11 (c) • Q11 (d) • Q11 (e) • Q12 (a) • Q12 (b) • Q13 (a) • Q13 (b) • Q13 (c) • Q13 (d) •
Question 05 (c) 

Suppose you are asked to find the point on closest to . Write down, but do not differentiate, an algebraic expression for the function that you wish to minimize. 
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 

To find the closest point on a curve from a given point is the same as to minimize the distance to that given point. 
Hint 2 

The distance between two points and is given by . 
Hint 3 

We can parametrize the curve by . 
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.

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. By Hint 3, any point on the curve can be written as . (Of course, this makes sense only for .) By Hint 1, we need to minimize the distance between and . By Hint 2, such distance is given by (In practice, one would minimize the square of the distance to simplify calculations, though.) Answer: We wish to minimize . 