MATH103 April 2017
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) (i) • Q1 (e) (ii) • Q1 (e) (iii) • Q1 (f) • Q2 (a) • Q2 (b) (i) • Q2 (b) (ii) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 • Q5 • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) • Q8 (c) • Q9 (a) • Q9 (b) • Q9 (c) • Q9 (d) • Q9 (e) •
[hide]Question 01 (d)
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For each graph (i) and (ii), identify the graph (A-D) which depicts its antiderivative and determine a consistent starting point of the integral.
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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!
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[show]Hint
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By the fundamental theorem of calculus, we have .
Also, notice that
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[show]Solution
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We observe that the functions in graphs (i) and (ii) are continuous, so by the fundamental theorem of calculus, we must have .
Let us first consider the function in graph (i). The function, , is negative when , positive when , and negative again when . Hence its antiderivative must be decreasing when , increasing when , and decreasing again when . This matches the function in graph B. To determine a starting point , we note that we must have and that the only clearly identifiable zero of the function in graph B is at . We conclude that the graph of in this case is B and that is a consistent starting point.
Next, let us consider the function in graph (ii). The function, , is negative when , positive when , and still positive when . Hence its antiderivative must be decreasing when , increasing when , and still increasing when . This matches the function in graph D. To determine a starting point , we note that we must have and that the only clearly identifiable zero of the function in graph D is at . We conclude that the graph of in this case is D and that is a consistent starting point.
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