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[answered] Calculus I (Math-UA-121.036) Fall 2016-HW 10 Section 4.5 1.

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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

x2

+ y2 = 1

9

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].

1

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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