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Practice Final Exam c

# Practice Final Exam c - b Where is the graph concave up...

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December 8, 1994 Math 1507 Final Exam NAME: Instructions: Write the answers where indicated and give clear evidence of your reasoning (or points will be taken oF). You may attach sheets with your work if it is organized enough to be helpful. Graphs should be clearly labeled. This exam is to be taken without the use of books, notes, calculators or any other such materials. All problems count 10 points I II III IV V VI VII VIII IX X Total 1

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I: Find the following derivatives : d dx tan ( x ) = d dx ( tan ( x ) + p sin ( x/ 3)) = d dx cos (2 /x ) = II: Compute the best linear ±t for the functions: a) f ( x ) = sinx + cosx at the value x = π/ 2. b) f ( x ) = x 1 / 2 + x 3 / 2 at the value x = 3. 2
III: Find the following derivatives: a) d dx f ( f ( f ( x ))), at x = 1, where f ( x ) = x 2 - 2. ANSWER: KEY FORMULA OR METHOD: b) d dx [ f ( f ( x )) · f ( x )], at x = 1, where f ( x ) = x 2 - 1. ANSWER: KEY FORMULA OR METHOD: IV: Compute the following limits, or state why they do not exist: a)lim x 0 sin (2 x ) cos ( x ) x = b)lim x 0 sin ( x 2 ) x 2 = 3

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V: Consider the function f ( x ) = x 3 - 6 x 2 + 12 x - 6. a) Where is the function increasing and where decreasing?

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Unformatted text preview: b) Where is the graph concave up, where concave down? Determine the inFection points. c) Sketch the graph below: VI: A jogger runs from P to Q (see ±gure). Along the street she is able to run 10 miles/hour and in the meadow she runs 6 miles/hour. What is her minimal time to run from P to Q? 4 VII: Sketch the the curve given by the polar representation r = 2 | cosθ | . VIII: Let z =-1-i and w = 2 + 3 i Find the following (in ”standard form). zw + 2 w = zw + wz = w/z = 5 IX: The graph below represents a twice diferentiable Function. In the spaces below, give a qualitative picture oF the ±rst and second derivative oF the Function. Clearly indicate where they are positive, zero or negative. X: Convert into polar coordinates: 5 + 10 i = 8-4 i = 6...
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Practice Final Exam c - b Where is the graph concave up...

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