Science:Math Exam Resources/Courses/MATH102/December 2013/Question A 05

MATH102 December 2013

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Question A 05

Consider the function $y=\cos(x)$ and the tangent line to this function at the point $x=\pi /2$. Using that tangent line as a linear approximation of the function would lead to (a) overestimating the value of the actual function for any nearby $x$. (b) underestimating the value of the actual function for any nearby $x$. (c) overestimating the function when $x>\pi /2$ and underestimating the function when $x<\pi /2$. (d) overestimating the function when $x<\pi /2$ and underestimating the function when $x>\pi /2$. (e) overestimating the function when $x<0$ and underestimating the function when $x>0$.

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 that over and under estimating the approximation is dependent on concavity. If the function is concave up, we have an underestimate. If it is concave down, it is an overestimate.

Hint 2
To check concavity, we need to look at the second derivative.

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. Taking the second derivative of our function gives ${\begin{aligned}y&=\cos(x)\\y'&=-\sin(x)\\y''&=-\cos(x)\end{aligned}}$ To the immediate left of $x=\pi /2$, the second derivative is negative and to its immediate right it is positive. So we get an overestimate when $x<\pi /2$ and an underestimate when $x>\pi /2$. Thus the answer is ${\color {blue}{\text{(D)}}}$.

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