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c1-t3m-a

# c1-t3m-a - NAME Brief Answers TEST-3M/MAC2311 Page 1 of 5...

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NAME: Brief Answers TEST-3M/MAC2311 Page 1 of 5 Read Me First: Show all essential work very neatly. Use correct notation when presenting your computations and arguments. Write using complete sentences. Be careful. Remember this: "=" denotes "equals" , " " denotes "implies" , and " " denotes "is equivalent to". Do not "box" your answers. Communicate. Show me all the magic on the page. 1. (10 pts.) (a) Using implicit differentiation, compute dy/dx and d 2 y/dx 2 when x 2 + y 2 = 25. Label your expressions correctly or else. d(x 2 + y 2 )/dx = d(25)/dx implies that dy/dx = -x/y. Consequently, after differentiating one more time using quotient rule, replacing the occurance of dy/dx in the expression for the second derivative, and cleaning up the algebra, we obtain d 2 y/dx 2 = -[y 2 + x 2 ]/y 3 . (b) Obtain an equation for the line tangent to the graph of x 2 + y 2 = 25 at the point (1,-(24) 1/2 ). Evaluating the implicit derivative above at (1,-(24) 1/2 ) provides us with the slope of the tangent line, namely 1/(24) 1/2 . Consequently, an equation for the tangent line at the desired point is y -(-(24) 1/2 ) = 1/(24) 1/2 (x - 1). 2. (5 pts.) A 5-ft. ladder is leaning against the wall. If the top of the ladder slips down the wall at a rate of 2 ft./sec., how fast will the foot be moving away from the wall when the top is 3 ft. above the ground? [Try 4.6: 12-15 1st!]

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