Tuesday, March 17, 2020
Stewartââ¬â¢s Calculus 8th Edition Section 1.1 Question 1
Stewartââ¬â¢s Calculus 8th Edition Section 1.1 Question 1 SAT / ACT Prep Online Guides and Tips This posts contains aTeaching Explanation. You can buyCalculus by Stewarthere. Why You Should Trust Me:Iââ¬â¢m Dr. Fred Zhang, and I have a bachelorââ¬â¢s degree in math from Harvard. Iââ¬â¢ve racked up hundreds and hundreds of hours of experienceworking withstudents from 5thgradethroughgraduate school, and Iââ¬â¢m passionate about teaching. Iââ¬â¢ve read the whole chapter of the text beforehand and spent a good amount of time thinking about what the best explanation is and what sort of solutions I would have wanted to see in the problem sets I assigned myself when I taught. Question:If$f(z) = z -âËÅ¡(2-z)$ and$g(u) = u -âËÅ¡(2-u)$is it true that f =g?Page in 8th Edition:19 Short Answer: Yes, it is true that f=g because the equation for g is exactly the same as that for f, except with x replaced by u. Homework Answer: Because the equation for f(x) and g(u) are the same, this means that for all valid inputs for function f, the function f and g give the same output. In other words, for all valid z, $f(z) = z - âËÅ¡(2-z) = g(z)$. Motivated Answer: This question is asking if f = g. What does it mean for two functions to be equal? We know that 2 = 2, and if someone asks, does 2=3? We know the answer is ââ¬Å"noâ⬠, but does f = g? Remember, functions take in inputs, and spit out outputs. Two functions f and g are only equal if they always give you the same output no matter what the input is. Letââ¬â¢s see what happens if we put in any valid input z into f. We get$f(z) = z - âËÅ¡(2-z)$. Now letââ¬â¢s put that same z into g, and we get$g(z) = z - âËÅ¡(2-z)$. These two are the same, and so f and g are the same. This question is a bit of a trick. The textbook writes$g(u) = u - âËÅ¡(2-u)$, but they could have just written$g(x) = x - âËÅ¡(2-x)$. This would have made it much more clear that f = g. There are two key learning points to take away: Two functions can be the same even if the equations look different written out. The above point is NOT true in reverse: If you substitute the same variable z into two functionsââ¬â¢ equations, and can get the equations to look the same, then the functions are the same. Video Solution: Get full textbook solutions for just $5/month. PrepScholar Solutions has step-by-step solutions that teach you critical concepts and help you ace your tests. With 1000+ top texts for math, science, physics, engineering, economics, and more, we cover all popular courses in the country, including Stewart's Calculus. Try a 7-day free trial to check it out.
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