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1st - x →-4 x<-4 so x 4< 0 lim x →-4-| x 4 | x 4 =...

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Calculus With Analytic Geometry Hakan TOR February 26,2010 1. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downwards. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. Solution: The answers are below, but for explanations they are not enough. (a) f(x)+3 (b) f(x)-3 (c) f(x-3) (d) f(x+3) (e) -f(x) (f) f(-x) (g) 3f(x) (h) f(x)/3 2. (a) What is wrong with the following equation? x 2 + x - 6 x - 2 = x + 3 (b) In view of part (a), explain why the equation lim x 2 x 2 + x - 6 x - 2 = lim x 2 ( x + 3) is correct. Solution: 1
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3. Use the Squeeze Theorem to show that lim x 0 p x 3 + x 2 sin π x = 0 Solution: Observe - p x 3 + x 2 p x 3 + x 2 sin π x p x 3 + x 2 since - 1 sin π x 1 x IR Since lim x 0 x 3 + x 2 = 0, lim x 0 x 3 + x 2 sin π x = 0 by the Squeeze Theorem. 4. Find lim x →- 4 - | x + 4 | x + 4 . Solution:
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Unformatted text preview: x → -4-, x <-4, so x + 4 < 0. lim x →-4-| x + 4 | x + 4 = lim x →-4--( x + 4) x + 4 = lim x →-4--1 =-1 5. *Prove that lim x → 2 1 x = 1 2 . Solution: 6. *If the function f is defined by ± if x is rational 1 if x is irrational prove that lim x → f ( x ) does not exist. Solution: 7. Suppose that lim x → a f ( x ) = ∞ and lim x → a g ( x ) = c , where c is a real number. Prove each statement. (a) lim x → a [ f ( x ) + g ( x )] = ∞ (b) lim x → a [ f ( x ) g ( x )] = ∞ if c > (c) lim x → a [ f ( x ) g ( x )] =-∞ if c < Solution: 8. From the graph of g, state the intervals on which g is continuous. Figure 1: Graph Solution: 9. Use the Intermediate Value Theorem to prove that there is a positive number c such that c 2 = 0. (This proves existence of the number √ 2) Solution: 2...
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