Science:Math Exam Resources/Courses/MATH103/April 2013/Question 09

MATH103 April 2013

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Question 09
Sketching Antiderivatives Consider the sketch $\displaystyle f(x)$ below. Provide a sketch of the antiderivative $\displaystyle A(x)=\int _{a}^{x}f(s)\,ds$ for $\displaystyle -1.5<x<1.5$. Make sure to clearly indicate local maxima and minima of $\displaystyle A(x)$

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 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
There are three key values to note - what happens at a and what happens at the two locations where the function goes from positive to negative?

Hint 2
Don't forget, if you start at a and go left you actually get more negative since integrals also encode a sense of direction.

Hint 3
Your function should be smooth (meaning differentiable everywhere - so no corners allowed!)

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. Attached is a picture. Notice that there is a local minimum at $\displaystyle x=-0.4$ and a local maximum at $\displaystyle x=-1.4$. Also note that plugging in a into the integral gives $\displaystyle A(a)=\int _{a}^{a}f(s)\,ds=0$.

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