01.21 - Week 1 worksheet for MATH 3283W PDF

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Week 1 worksheet for MATH 3283W...

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MATH 3283W Worksheet 1 answers Tuesday 21 January 2020

Sentences, not statements Many answers. For example: “I am up for the challenge!” You have been writing mathematical statements for a long time! Many answers. For example: 1. 10 × 10 = 100. 2. 1 + 1 = 3. Note that the equal sign is representing the verb in these statements! 3. The equation x2 + 1 = 0 has no real solutions. 4. There is a real number that satisfies the equation x2 + 1 = 0. 5. The derivative of the function f (x) = x2 is f ′ (x) = 2x. 6. The series 1 +

1 2

+

1 3

+ 41 + · · · converges. The natural numbers

1. A natural number n is odd if there exists a natural number k such that n = 2k − 1. 2. A natural number n is composite if there exist natural numbers p, q > 1 such that n = p · q . 3. Many answers. For example: there is a natural number that is prime and not odd. 4. Many answers. For example: there is a natural number that is odd whose square is prime. The real numbers and some calculus 1. The value of the integral is 4, the area of a trapezoid of base 1 and perpendicular sides 3 and 5. R1 2. 0 (2x + 3) dx = [x2 + 3x]01 = 4.

3. See Lay pages 293 and 295, or see Stewart (custom 7th edition) pages 388 and 391. Here we used part II. Note that we will not cover chapters 6 and 7 of the Lay book in this course. 4. See Lay page 237. In this definition, x is changing. Or, see Stewart page 146. In this definition, h is changing. 5. See Stewart page 372. In this definition, n is changing; the definite integral is the limit of a sequence, that is, a function of n, the number of subdivisions of the interval [a, b]. Contrast Lay page 276, where the definite integral is defined in terms of infimum and supremum, two concepts that we will explore in this course (if not this definition of integral itself). 6. Z

0

1

n   X i (2x + 3) dx = lim 2 n→∞ n i=1



  1 +3 = lim n→∞ n

2 n2

n X i=1

!

i +3

!

  1 1 + + 3 = 4. n→∞ n

= lim

(Note that if a function is integrable like this one, then the value is the same no matter how the sample inputs are chosen on the subintervals.)...