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Derivative as a Rate of Change
Problem 153: When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time (hours) was . Find the growth rates at a. hours. b. hours. c. hours. Solution:…
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Perimeter of an Isosceles Triangle
Problem 152: The perimeter of an enclosed isosceles triangle is 40 feet. The length of the shortest side is 7 less than half of the two longest (equal) sides. What are the length of the sides of the enclosure? Solution: The perimeter of an isosceles triangle is given by (1) where is the perimeter,…
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Perimeter of a Rectangle
Problem 151: A rectangle garden has a length that is 10 feet longer than its width. The perimeter is 42 feet. Wat are the dimensions of the garden? Solution: The perimeter of a rectangle is given by (1) where is the perimeter, is the length of the sides, and is the width. We are…
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Slant Asymptote
Problem 150: Find the slant asymptote of the following function: (1) Solution: In order to find the slant asymptote of a function, we need to use long division. That is, using long division we get (2) where 1 is the remainder and gives us the slant asymptotes. The slant asymptote is given by…
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Rates of Changes
Problem 149: An object is dropped from the top of a 100-m-high tower. Its height above the ground after sec is m. How fast is it falling 2 sec after it is dropped? Solution: We will let be the height above the ground after sec. That is, . Since we are interested on how fast…
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Limit of the form Sin(x)/x
Problem 148: Compute the following limit: (1) Solution: In order to solve this problem, we will use two different formulas. (2) That is, (3) Therefore, (4)
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Limit Laws II
Problem 147: (a) If , find . (b) If , find . Solution: We need to use the limit laws to solve this problems. (a) (1) That is, (2) (b) (3) That is, (4)
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Limit Laws
Problem 146: If , find . Solution: We will use the limit laws to solve this problem. (1) That is, (2)
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Continuous Function II
Problem 145: At what point is the following function continuous? (1) Solution: One can think of continuity as where is the function (1) no defined. That is, we need to find the domain of the function in order to come to this conclusion. That is, (2) Note that we do not want to…
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Continuous Functions
Problem 144: At what point is the following function continuous? (1) Solution: One can think of continuity as where is the function (1) no defined. That is, we need to find the domain of the function in order to come to this conclusion. That is, (2) Note that we do not want to…
One Problem at a Time
