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Derivatives in Mathematics and Physics
Problem 173: A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the rectangle can have, and what are its dimensions? Solution: Let be the coordinates of the corner of the rectangle obtained by placing the circle and rectangle in the coordinate plane. The length, height, and area…
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Derivatives in Biology
Problem 172: Find the amount of medicine to which the body is most sensitive by finding the value of that maximizes the derivative , when (1) and is a constant. Solution: We need to find the derivative with respect to twice. That is, (2) That is, by the Second Derivative Test, gives the…
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Derivatives in Business and Economics
Problem 171: It costs you dollars each to manufacture and distribute backpacks. If the backpacks sell at dollars each, the number sold is given by (1) where and are positive constants. What selling price will bring a maximum profit? Solution: In order to find the maximum profit, we need to find an equation for…
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Derivatives in Physics
Problem 170: The height above ground of an object moving vertically is given by (1) with in feet and in seconds. Find a. the object’s velocity when b. its maximum height and when it occurs; c. its velocity when . Solution: a) The object’s velocity is the derivative of the position. That is, when…
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Derivatives in Applications
Problem 169: A rectangle plot of farmland will be bounded on one side by river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? Solution: Let be the length of the rectangle…
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Linearization of a Function at a Point
Problem 168: Find the linearization of at : (1) Solution: The linearization of a differentiable function at a point is given by (2) Hence, (3) Therefore, (4) That is, (5)
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Linearization of a Function at a Point
Problem 167: Find the linearization of at : (1) Solution: The linearization of a differentiable function at a point is given by (2) Hence, (3) Therefore, (4) That is, (5)
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Related Rates IV
Problem 166: Two airplanes are flying in the air at the same height: airplane is flying east at 250 miles per hour and airplane is flying north at 300 miles per hour. If they are both heading to the same airport, located 30 miles east of airplane and 40 miles north of airplane , at…
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Related Rates III
Problem 165: A 10ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2ft per sec, how fast is the bottom of the ladder moving away from the wall when the bottom of the ladder is 5ft from the wall? Solution: The ladder and…
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Related Rates Problem II
Problem 164: A hot air ballon rising straight up from a level field is tracked by a range finder 150 m from the liftoff point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of rad/min. How fast is the ballon rising at that moment? Solution: When…
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