Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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 a "approaches" to a (symbolically: a → a) • Ar is the signed distance between a and x: Ax = x – a • signed distance just means that Ax is negative when x 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 = – a is the same as x = a + Ax • replacing x with a + Ax, our point B is instead: (a + Ax, f(a + Ax)) • 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 Ar-0 Ar • 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 + Δε) -f(a) • so lim Ar0 is written more simply: Δε 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 rational 1.0 f(x) = 2, compute the instantaneous slope at: (7, ?) Start with two points A : (7,) and B : (7+ h, ?) What is the y-coordinate of B?
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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 a "approaches" to a (symbolically: a → a) • Ar is the signed distance between a and x: Ax = x – a • signed distance just means that Ax is negative when x 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 = – a is the same as x = a + Ax • replacing x with a + Ax, our point B is instead: (a + Ax, f(a + Ax)) • 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 Ar-0 Ar • 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 + Δε) -f(a) • so lim Ar0 is written more simply: Δε 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 rational 1.0 f(x) = 2, compute the instantaneous slope at: (7, ?) Start with two points A : (7,) and B : (7+ h, ?) What is the y-coordinate of B?
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