Homework 2-solutions

# Homework 2-solutions - saliyev(is4663 – Homework 2 –...

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Unformatted text preview: saliyev (is4663) – Homework 2 – knopf – (55420) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. This assignment covers Sections 2.4, 2.5, 3.1, and 3.2. 001 10.0 points How close to − 5 do we have to take x for the inequality 1 ( x + 5) 2 > 700 to hold? 1. within at least 0 . 037 correct 2. within at least 0 . 077 3. within at least − . 003 4. within at least 0 . 017 5. within at least 0 . 057 Explanation: We have to find the largest value of δ so that | x + 5 | < δ = ⇒ 1 ( x + 5) 2 > 700 , or, in other words, so that | x + 5 | < δ = ⇒ ( x + 5) 2 < 1 700 . Thus x must satisfy the inequality | x + 5 | < 1 √ 700 = 0 . 037 . Consequently, x must be within at least 0 . 037 of − 5 if the inequality 1 ( x + 5) 2 > 700 is to hold. 002 10.0 points Determine which of the following could be the graph of f near the origin when f ( x ) = x 2 − 7 x + 10 2 − x , x negationslash = 2 , 4 , x = 2 . 1. 2. correct 3. saliyev (is4663) – Homework 2 – knopf – (55420) 2 4. 5. 6. Explanation: Since x 2 − 7 x + 10 2 − x = ( x − 2)( x + 5) 2 − x = 5 − x , for x negationslash = 2, we see that f is linear on ( −∞ , 2) uniondisplay (2 , ∞ ) , while lim x → 2 f ( x ) = 3 negationslash = f (2) . Thus the graph of f will be a straight line of slope − 1, having a hole at x = 2. This eliminates four of the possible graphs. But the two remaining graphs are the same except that in one f (2) > lim x → 2 f ( x ) , while in the other f (2) < lim x → 2 f ( x ) . Consequently, must be the graph of f near the origin. 003 10.0 points Determine which (if any) of the following functions is not continuous at x = 7. 1. f ( x ) = braceleftBigg 1 x − 7 x negationslash = 7 7 x = 7 correct 2. f ( x ) = 1 | x − 5 | x ≥ 7 1 2 x < 7 3. f ( x ) = braceleftBigg 28 2 x − 7 x negationslash = 7 4 x = 7 4. f ( x ) = 1 x − 5 x ≥ 7 1 2 x < 7 5. all continuous at x = 7 6. f ( x ) = braceleftbigg | x − 7 | x negationslash = 7 x = 7 saliyev (is4663) – Homework 2 – knopf – (55420) 3 Explanation: A function f will be continuous at x = 7 when f (7) exists and lim x → 7 f ( x ) = f (7) . Now f (7) exists for all the functions defined above; in addition, inspection shows that all these functions have the property lim x → 7 f ( x ) = f (7) except for f ( x ) = braceleftBigg 1 x − 7 x negationslash = 7 7 x = 7 . Consequently, this function is the only one that is not continuous at x = 7. 004 10.0 points Find all values of x at which f ( x ) = 5 1 − sin x fails to be continuous. 1. x = 2 nπ + 3 π 2 , all integers n 2. x = nπ + π 2 , all integers n 3. x = nπ + π 4 , all integers n 4. x = nπ + 3 π 4 , all integers n 5. x = nπ, all integers n 6. x = 2 nπ + π 2 , all integers n correct 7. x = 2 nπ, all integers n 8. x = (2 n + 1) π, all integers n Explanation: The only values of x...
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Homework 2-solutions - saliyev(is4663 – Homework 2 –...

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