18.01 PROBLEM SET 1

Hand in to Room 2-108 by 1:00 pm Wednesday, February 14.

The due date and time are absolute. Late homeworks are accepted until the solutions are posted, but the score will be multiplied by 1/2. Everything included here may be used later. You do not need to

memorize the proofs, but you need to know the facts.

1. G.10/1, 4, 5, 6. 34/4, 14, 20, 32, 43, 48.

(10 points)

2. Use the following formula:

(a + b)^{5} = a^{5} + 5a^{4}b + 10a^{3}b^{2} + 10a^{2}b^{3} + 5ab^{4} + b^{5},

to prove that

(15 + \sqrt{220})^{5} + (15 − \sqrt{220})^{5}

is an integer, divisible by 10. What is the last digit preceding the decimal point in the decimal expression of (15 + \sqrt{220})^{5}?

(3 points)

3. Using the notation of the picture below prove that t = tan θ and s = sec θ. Find similar statements for cot θ and csc θ. (This exercise shows the names ’tangent’ and ’secant come from.)

(3 points)

4. Using the above picture prove that sin θ < θ < tan θ.

(3 points)

5. Find lim_{θ→0} \frac{sin θ}{θ} .

(3 points)

6. Find a function that is not continuous at any point.

(3 points)

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