Name ____________________________ Math 50A – Unit 3 Exam (“Take Home”) – Due Tues July 21 11 problems, 110 Points Total (10 points each) You may use pencil, eraser, graph paper, plain paper, scratch paper, and your calculator. With scratch paper, (a) write your name on it, (b) number each problem, and (c) turn it in with the test. Read Carefully. Show all relevant work for full credit. NOTE: Please do your work on separate paper. Number the problems; you NEED NOT re-­‐write the problems. Show all relevant work for full credit. Indicate answer clearly. 1. Use calculus to determine where f ( x) = x 3 − 2x 2 − 3x + 4 is increasing and where it is decreasing. Show all relevant work. 2. Let f ( x) = x 3 − 2x 2 − 3x
€ + 4 a. List the intervals over which is concave up and concave down. b. List any inflection points. €
Show all relevant work. 3. Use calculus to find all local and absolute extrema of the following: a. f ( x ) = 10 + 27x − x 3 on the interval [0, 4]. b. f ( x ) = 2x + sin x on [0, 2π] c. f ( x) = x 3 − 2x 2 − 3x + 4 € Show all relevant work. €
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4. Use the Intermediate Value Theorem to find an interval [a, b] for which f ( x) = x 4 − 3x 2 − 4x + 2 has at least one real root. Show all relevant work. 5. Determine whether each limit is an indeterminate form. If so, €
give the form and find the limit. Show all relevant work. a.
b.
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c.
x 2 − 7x +12
lim
x →3
x2 − 9
sin x
lim
x →0 x
ln x
lim
x →∞ 3x
6. A bacteria population is growing in such a way that after t hours, its population is 1000 + 10t + t3. Find the growth rate after 3 hours. Show all relevant work. 7. A circle is increasing in area. Find the rate of change of the area with respect to the radius r. Show all relevant work. 8. Air is being pumped into a chamber so that after t seconds the pressure in the chamber is 3 + 3t + t2 pounds/in2. Find the rate of change of the pressure at t=2 seconds. Show all relevant work. 9. A balloon is rising at a constant speed of 5 ft/s. A cyclist is travelling along a straight road at a speed of 15 ft/s. When the cyclist passes under the balloon it is 45 feet directly above. How fast is the distance between the cyclist and the balloon increasing 3 seconds later? Show all relevant work for full credit, including at least one clearly labeled diagram. 10. Find the point on the hyperbola that is closest to the point (3,0). Show all relevant work. 11. Sketch the graph of a function that satisfies all the given conditions: a.
Domain: ( −∞,0) U(0, ∞ ) b.
c.
€ d.
f (−1) = 0 lim f ( x) = ∞ €x →0 +
lim f ( x) = −∞ x →0 −
( 21 , ∞) € e.
f ' ( x) > 0 on € f.
f ' ( x) < 0 on ( −∞,0) U 0, 21 € g.
f"( x)
€ > 0 on ( −∞, −1) U(0, ∞) € h.
f"( x) < 0 on ( −1,0) €
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( )
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BONUS: The triangle in the figure below has, as its three sides, the x-­‐axis, the y-­‐axis, and a line that goes through the point (3,2). y
(a)
Is there a minimal area for such a triangle? If so, what is it? If not, why not? (3,2)
(b)
Is there a maximal area for such a triangle? If so, what is it? If not, why not? x