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PFBSolns

# PFBSolns - MATH 31A(Butler Practice for Final(B Try to...

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MATH 31A (Butler) Practice for Final (B) Try to answer the following questions without the use of book, notes or calculator; but you can use the equation sheet posted on the course website. Time yourself and try to finish the questions in less than three hours. 1. Show that Z π/ 4 0 tan x dx + Z 1 0 arctan x dx = π 4 . (Hint: interpret each integral as an area and show how the areas “piece” together.) First we note that we can change the “dummy” variables inside the integral. In particular, we are trying to find Z π/ 4 0 tan y dy + Z 1 0 arctan x dx. But since x = tan y if and only if y = arctan x (for x and y in this range), then these two integrals both involve the same curve. In particular, these integrals correspond to the respective areas shown below. 1 π 4 y =arctan x or x =tan y R 1 0 arctan x dx R π/ 4 0 tan y dy Since the two areas piece together to form a rectangle with dimensions 1 and π 4 then the combined area is the area of the rectangle which is π 4 .

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2. A particle moves along the curve implicitly defined by xy 4 - yx 4 = x - y 2 . When the particle passes through the point (1 , 1) its x coordinate is changing 1 / 4 units per second. How fast is the y coordinate changing? We have x = 1, y = 1 and dx dt = 1 4 and we are looking for dy dt . This is a related rates problem and so we take the derivative of both sides of the implicitly defined function with respect to t . So we have dx dt · y 4 + x · 4 y 3 dy dt - dy dt · x 4 - y · 4 x 3 dx dt = dx dt - 2 y dy dt . Substituting what we know we have 1 4 · 1 4 + 1 · 4 · 1 3 dy dt - dy dt · 1 4 - 1 · 4 · 1 3 · 1 4 = 1 4 - 2 · 1 · dy dt , which rearranges to (4 - 1 + 2) dy dt = 1 4 - 1 4 + 1 or 5 dy dt = 1 or dy dt = 1 5 units sec .
3. The night before the final you have decided to do one last study session. But before you begin you decide that you want to make the best use of your time. You know that there is a diminishing return to the amount of time you study (i.e., you get more out of your first hour of study than you will your second; and more out of your second hour than you will your third). At the same time you know that the longer you study the less sleep you will have and the harder it will be to concentrate on the test (which will make problems about optimizing your study session even harder!).

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PFBSolns - MATH 31A(Butler Practice for Final(B Try to...

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