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Calculus I (Math-UA-121.036) Fall 2016-HW 10 Section 4.5
1. There are 50 apple trees in an orchard. Each tree produces 800 apples. For each
additional tree planted in the orchard, the output per tree drops by 10 apples. How
many trees should be added to the existing orchard in order to maximize the total
output of trees ?
2. Find the point(s) on the ellipse
+ y2 = 1
closest to the point (2, 0).
[Hint: Minimize the square of the distance?this avoids square roots.]
3. A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance
x from the front of the room should you position yourself so that the viewing angle ?
of the movie screen is as large as possible ?
4. A cylinder with open top with radius r and height h has surface area 8 cm2 . Find the
largest possible volume of such a cylinder?
5. The hypotenuse of a right triangle has one end at the origin and one end on the curve
y = x2 e?3x , with x ? 0. One of the other two sides is on the x-axis, the other side
is parallel to the y-axis. Find the maximum area of such a triangle. At what x-value
does it occur? Section 5.1
6. Recall that Rn denotes the right-endpoint approximation using n rectangles, Ln denotes
the left-endpoint approximation using n rectangles, and Mn denotes the midpoint
approximation using n rectangles.
Calculate the approximation for each of the given function and interval below.
(a) R5 ,
(b) L6 ,
(c) M4 , f (x) = x2 + x on the interval [?1, 1].
f (x) = 6x + 2 on the interval [1, 3].
f (x) = 2
on the interval [1, 5].
x +1 7. The figure below shows the velocity of a particle, in cm/sec, along the t-axis for ?3 ?
t ? 3. 1 Calculus I (Math-UA-121.036) Fall 2016-HW 10 (a) Describe the motion in words: Is the particle changing direction or always moving
in the same direction? Is the particle speeding up or slowing down?
(b) Find an overestimate and an underestimate of the distance travelled over the time
period t = ?3 to t = 3. You could use any number of rectangles (but at least 3)
that you deem appropriate. 2
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