Math 180 -Assignment

Math 180 -Assignment #4 Due: 10:00a, Monday, March 29th, 2021 ************************************************************************** READ AND FOLLOW DIRECTIONS **Complete solutions using proper notation will be required to receive full credit. *Please clearly identify your final answers. Circle your final answers where appropriate. **Please start each NUMBERED PROBLEM on a new page. *Please put your problems in numerical order before submitting them as a single PDF or Word Document via Blackboard. **Do NOT wait until the last minute. Late submissions will receive severe reductions. *Each numbered problem will be graded on a 10-point scale. The total will be scaled to 100 points. ***************************************************************************** #1.) Choose A or B, but not both. A.) Use differentials to approximate 79 . Hint: f ( x ) = x , x = 81 B.) Use differentials to approximate the change in f ( x ) = x + 7 as the value of x changes from x = 9 to x = 9.05 #2.) Choose A.) or B.), but NOT both. To receive full credit you will need to provide a sketch/diagram with correct labels. A.) Consider the rectangle inscribed inside the region bounded by the graph of 1 f ( x ) = − x 2 + 12 and the x-axis. Find the dimensions of the rectangle with maximum area. 3 Find the maximum area of this rectangle. B.) Consider the rectangle inscribed inside the region bounded by the graphs of 1 1 y = x + 9, y = − x + 9, and the x-axis. Find the dimensions of the rectangle with 3 3 maximum area. Find the maximum area of this rectangle. #3.) You measure the circumference of a circle, C = 62.8, and know the measurement is accurate to within 0.05 inches. You use this measurement to calculate the radius of the circle. A.) Use differentials to approximate the Maximum Propagated Error in Calculating the Radius. B.) Determine the Maximum Percent Error in Calculating the Radius. Note: You will need to write the radius as a function of the circumference to start. #4.) Use Newton’s Method to find the all values of x where the graphs of the given 1 3 functions intersect. g ( x ) = x3 + x 2 − 5 x −12 and h ( x ) = − x. 2 2 This problem should be worked using a spreadsheet. I will provided a template for Newton’s method on Blackboard. I suggest you use the template and print using landscape setting. Please note: When evaluating integrals, there might be one that you can not evaluate because you do not know an antiderivative. #5.) Evaluate each integral. Please demonstrate the properties by showing the steps. a.)  (3x − sin x) dx b.)  (4 x − 5sec x  tan x ) dx #6.) Evaluate each integral. Please demonstrate the properties by showing the steps. a.)  3csc x dx b.)  1 4   2  − cos  d   #7.) Evaluate each integral. Please demonstrate the properties by showing the steps. a.)  4 y5 −16 y 4 − 24 y 2 dy 8 y4 b.) 12×4 (4×3 −15) dx #8.) Evaluate each integral. Please demonstrate the properties by showing the steps. a.)  3w2 − 4 w dw 3w b.)  3csc x ( 2csc x + 5sec x ) dx 1 #9.) Choose A.) or B.), but NOT both. Given f ( x ) = − x2 + 12,  0,6  , 3 A.) Find the Lower Sum. B.) Find the Upper Sum. Extra Credit.) Consider an inscribed rectangle in the region bounded by the graphs of 1 i.) y = x + 6,  −18, 0 , 3 1 ii.) y = − x 2 + 6,  0, 6 6 iii.) the x-axis. Find the dimensions of the rectangle with maximum area. Find the maximum area of this rectangle.
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