Test 1 - a ) lim x 1 x 2 + 2 x-3 x 2 + x-2 b ) lim x 1 2 x...

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Test I for Calculus I, Math 1501C1, September 10, 2002 Name: This test is to be taken without calculators and notes of any sorts. The allowed time is 50 minutes. Write answers in boxes where provided. Provide exact answers; not decimal approximations! For example, if you mean 2 do not write 1 . 414 . . . .
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I: (25 points, No partial credit) a) For some function f ( x ) you know that | f (1 + h ) - f (1) | ≤ 5 | h | 2 . Find δ > 0 so that | f (1 + h ) - f (1) | < ε whenver | h | < δ . b) True or false a < b a 2 < b 2 0 < a < b a < b c) Find all x for which x 2 + x - 2 > 0 d) Find the linear function ( x ) whose slope is 2 and has the point (2 , 1) on its graph. e) Find the linear function ( x ) that has the points (2 , 1) and (3 , 2) on its graph.
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II: (25 points) Which of the following limits exist? Compute them if they exist. Otherwise explain why they do not exist.
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Unformatted text preview: a ) lim x 1 x 2 + 2 x-3 x 2 + x-2 b ) lim x 1 2 x 2 + 1 3 x-1 c ) lim x 1-cos(sin( x )) x 2 d ) lim x 1 p ( x-1) 2 + 1-1 x-1 e ) lim x | sin( x ) | x III: (25 points) Given the function f ( x ) = 5 9 + x 2 . a) Find all values for x that satisfy | f ( x )-f (4) | &lt; 1 b) Find g ( h ) so that | f (4 + h )-f (4) | = g ( h ) | h | c) For &gt; 0, nd &gt; 0 so that | h | &lt; | f (4 + h )-f (4) | &lt; IV: (25 points) Show, using Mathematical Induction, that for n = 1 , 2 , . . . 7 n-6 n-1 is divisible by 36. Fill in the following steps: a) Check the statement for n = 1 and n = 2. b) Write down the Induction Assumption: c) Carry out the Induction Step:...
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Test 1 - a ) lim x 1 x 2 + 2 x-3 x 2 + x-2 b ) lim x 1 2 x...

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