Tutor profile: Elliott G.
Explain in your own words the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus, in my opinion, binds calculus together. It shows that if we take the derivative of a function, and then integrate it, we get the original function. For example, the derivative of x^2 is 2x. When we integrate 2x, we get x^2 + C. The converse is also true. If you integrate a function, and then take its derivative, you'll end up with the original function. For example, if we integrate 5x^4, we get get x^5. If we take the derivative of x^5, we get 5x^4. The Fundamental Theorem of Calculus shows there's a link in mathematics between area and rate of change.
Domestic kittens at nine weeks weigh an average of 24.5 oz. Kitten weights are normally distributed with a standard deviation of 5.25 oz. Find the probability that a kitten will weigh less than 14 oz.
What is this problem asking? Let's look at the context. Kittens weigh an average of 24.5 oz., but it's highly likely that a kitten's not going to weigh that exact amount. Weights of kittens are "normally distributed" over a "bell curve," where most kittens weigh close to 24.5 oz, but a few weigh a lot more, or a lot less. A kitten that only weighs 14 oz seems far below average. How far? Let's find the z-score: z = (x - µ) / s x = 14, the random variable µ = 24.5, or the mean s = 5.25, or the standard deviation z = (14 - 24.5) / 5.25 z = -2 Sure enough, this kitten weighs 2 "standard deviations" below average. One could infer this kitten is pretty underweight. To find the probability, we use a "z-table," which can be found online. When we look up the value for -2, we get 0.0228. There is a 2.28% chance a kitten will weigh less than 14 oz.
A class grade is determined by four tests worth 20% each, a 10% homework grade, and a 10% classwork grade. Jamaal received a 90% homework grade, a 95% classwork grade, and three test scores of 98%, 72%, and 93%. What is the minimum score he'd need in order to get a 90% average in the class?
In the problem, we know all of Jamaal's grades except for one: a test worth 20% of his grade. We want his weighted average to add up to 90%, or 0.90. When finding a weighted average, multiply each score (in decimal form) by its weight and add them together. We'll define the score Jamaal needs to get an A- as x. This means that: (0.1 * 0.9) + (0.1 * 0.95) + (0.2 * 0.98) + (0.2 * 0.72) + (0.2 * 0.93) + 0.2x = 0.90 Note that, since homework is work less than tests, we multiply his 90% homework grade by 0.1, while we multiply his 88% test grade by 0.2. Following order of operations and then solving for x, we get: 0.09 + 0.095 + 0.196 + 0.144 + 0.186 + 0.2x = 0.90 0.705 + 0.2x = 0.9 -0.705 -0.705 0.2x = 0.195 x = 0.975 Jamaal needs a 97.5% on his last test to get an A- in the class.
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