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when f(x) is polynomial 44,8) Fie -16 f(x) = }x² – 5x + 26 compute the instantaneous slope at: (4, 8) art with two points : (4, 8) and B : (4 + h, ?) hat is the y-coordinate of B?

when f(x) is polynomial 44,8) Fie -16 f(x) = }x² – 5x + 26 compute the instantaneous slope at: (4, 8) art with two points : (4, 8) and B : (4 + h, ?) hat is the y-coordinate of B?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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• slope requires two points, one of which is always the point where we want to find the instantaneous slope. • A: (a, f(a)) where a is a specific real number (the x-coordinate where we want to find the instantaneous slope) • the second point varies, and we look at the slope as this point "gets close" to A. • B: (x, f(x)) where x "approaches" to a (symbolically: x → a) • Ar is the signed distance between a and x: Ax = x - a • signed distance just means that Ax is negative when a approaches a from the left x → a , and Ax is positive when x approaches a from the right x + a" • as x approaches a, this distance gets closer and closer to 0 • in other words, x → a has the same meaning as Ax → 0 • Flip the script: Ax = x - a is the same as x = a + Ax • replacing a with a + Ax, our point B is instead: (a + Ax, f(a + Aæ)) • and Ay = f(x) – f(a) becomes Ay = f(a + Ax) – f(a). • also, instead of x → a, we have Ax - 0 representing the same idea Ay • this means lim Ay will instead be written as: lim Ar0 Ax • This adjustment will make our calculations easier in some cases. • Ar is cumbersome to write, so we frequently use h instead of Ax. f (a + Δ )-f (a) • so lim Ar-0 is written more simply: Ar f(a + h) – f(a) lim h-0 h • the benefit of this approach is that our denominator is now a monomial, which tends to be easier to reduce. The instantaneous slope of f(x) when f(x) is polynomial 4,8) F10 f(x) = x? – 5x + 26 compute the instantaneous slope at: (4, 8) Start with two points A : (4,8) and B : (4 + h, ?) What is the y-coordinate of B?
Transcribed Image Text:• slope requires two points, one of which is always the point where we want to find the instantaneous slope. • A: (a, f(a)) where a is a specific real number (the x-coordinate where we want to find the instantaneous slope) • the second point varies, and we look at the slope as this point "gets close" to A. • B: (x, f(x)) where x "approaches" to a (symbolically: x → a) • Ar is the signed distance between a and x: Ax = x - a • signed distance just means that Ax is negative when a approaches a from the left x → a , and Ax is positive when x approaches a from the right x + a" • as x approaches a, this distance gets closer and closer to 0 • in other words, x → a has the same meaning as Ax → 0 • Flip the script: Ax = x - a is the same as x = a + Ax • replacing a with a + Ax, our point B is instead: (a + Ax, f(a + Aæ)) • and Ay = f(x) – f(a) becomes Ay = f(a + Ax) – f(a). • also, instead of x → a, we have Ax - 0 representing the same idea Ay • this means lim Ay will instead be written as: lim Ar0 Ax • This adjustment will make our calculations easier in some cases. • Ar is cumbersome to write, so we frequently use h instead of Ax. f (a + Δ )-f (a) • so lim Ar-0 is written more simply: Ar f(a + h) – f(a) lim h-0 h • the benefit of this approach is that our denominator is now a monomial, which tends to be easier to reduce. The instantaneous slope of f(x) when f(x) is polynomial 4,8) F10 f(x) = x? – 5x + 26 compute the instantaneous slope at: (4, 8) Start with two points A : (4,8) and B : (4 + h, ?) What is the y-coordinate of B?
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