Calculus with Vectors by Jay S. Treiman (auth.)

By Jay S. Treiman (auth.)

Calculus with Vectors grew out of a robust want for a starting calculus textbook for undergraduates who intend to pursue careers in STEM fields. The technique introduces vector-valued services from the beginning, emphasizing the connections among one-variable and multi-variable calculus. The textual content comprises early vectors and early transcendentals and contains a rigorous yet casual method of vectors. Examples and targeted purposes are good awarded in addition to an abundance of motivating exercises.

The ways taken to themes comparable to the derivation of the derivatives of sine and cosine, the method of limits and using "tables" of integration were converted from the criteria noticeable in different textbooks which will maximize the convenience with which scholars may perhaps understand the fabric. also, the cloth offered is deliberately non-specific to any software program or platform so one can accommodate the wide range and swift evolution of instruments used. know-how is referenced within the textual content and is needed for quite a few problems.

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Example text

Using the theorem on limits above, we can prove the following result. The proof is left to the ambitious reader. 2 Limits and Continuity 45 Theorem 19 (Continuous vector valued functions, rules). Let f(x), g(x), r(x) and h(x) be functions that are continuous at a. Assume that h(a) = 0. Then the following hold: (i) f + g is continuous at a, (ii) r f is continuous at a, (iii) f · g is continuous at a, and f is continuous at a. (iv) h This theorem can be used to demonstrate that many elementary functions are continuous on their domains.

N→∞ Exercises 1. Find the limits of the following sequences as n goes to infinity. (a) an = (b) an = (c) 2. bn = an = (b) an = (c) an = n bn = (e) cn = n3/2 n2 √2 n (f) cn = (−1)n n3 2 n+5 n n−5 n−10 n+6 (d) an = (e) an = (f) an = n2 +n−1 n3 n2 +n−1 10−n+6n2 n2 +n−1 6−n Find the limits of the following sequences as n goes to infinity. (a) an = (b) an = (c) an = 4. n3/2 (d) Find the limits of the following sequences as n goes to infinity. (a) 3. 2 n 2 n2 2 n+5 + n n−5 + n−10 n+6 1 n2 3n 2n+2 4 n (d) an = (e) an = (f) an = n2 +n−1 6n+1 n−5 2n2 n2 +n−1 n2 + 100n−1 10−n+6n2 n2 +n−1 10n 6−n 3n2 +4 Explain why the following sequences do not converge as n goes to infinity.

If L ≥ 0 then, for odd n the inequality |an −L| = |−1−L| ≥ 1 > 1/2 holds. Similarly, if L < 0, for even n we have |an − L| ≥ 1 > 1/2. See Fig. 2a on page 35. This means that the sequence never stays near a given L, and the limit does not exist. Now that the idea of a limit for sequence has been defined, the sequence in Zeno’s paradox can be examined more closely. Recall that the sequence is {(1/2n, 1 − 1/2n)}. See Fig. 3 on page 35. The distance of the nth term, a(n), in the sequence from (0, 1) is √ 1/((2n )2 ) + 1/((2n)2 ) = 2/2n .

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