A farmer has produced 1000 apples and wants to sell them. He can sell the apples at two different markets:
- At market 1, if the farmer sells x apples, he can sell them for 2/square root of x dollars each
- At market 2, if the farmer sells y apples, he can sell them for 4/square root of y dollars each.
Find out how the farmer should split his 1000 apples between both markets in order to maximize profit. (Use Optimization)
In: Math
Please show work.
1. Find the derivatives dy/dx and d2y/dx and evaluate them at t = 2.
x = t2, y = tln t
2. Find the arc length of the curve on the given interval
x = ln t, y = t + 1, 1 ≤ t ≤ 2
3. Find the area of region bounded by the polar curve on the given interval.
r = tan θ, π/6 ≤ θ ≤ π/3
4. Find the length of the polar curve on the given interval. (12 points)
r = θ, 0 ≤ θ ≤ π/2
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For the given function determine the following: f (x) = (sin x + cos x) 2 ; [−π,π] a) Find the intervals where f(x) is increasing, and decreasing b) Find the intervals where f(x) is concave up, and concave down c) Find the x-coordinate of all inflection points
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1. An astronaut with a spaceship moves from left to right following the curve y = x^2. When he turns off the engine, the spacecraft will move along the tangent line at the point where he is at that moment. At what point does he have to turn off the engine to reach the point (4,15)?
6) There are two tangents to the function graph y = 4x − x^2 through the point (2,5). Determine the tangent equation. Hint: Suppose the contact point is (x0, y0), get two conditions that must be met by that point.
7) An ant goes from left to right along the top of the function graph y = 7 − x^2. A spider waits for ants at point (4, 0). Look for the distance between ants and spiders the first time they see each other.
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he organizers of a Reindeer Exhibit have asked you to build temporary yards for two reindeer; each yard will be surrounded by candy-cane fencing on all sides. The geometrically-minded reindeer only like to stay in square and circular yards; they can stay either together or separately. The organizers prefer not to build two yards of the same shape. Your task is to design yards that satisfy both the organizers and the reindeer: you must use the fencing given to build 1 or 2 yards that are circular or square (if there are two yards they must be different shapes).
The reindeer are most happy when the combined enclosed area of the yard(s) is the greatest possible. You have exactly 150 feet of candy-cane fencing. How will you design the yard(s) to optimize the reindeer’s happiness, under the constraints given?
(a) (3 points) In this problem, do you want to maximize or minimize an area?
(b) (4 points) Let be the length of the side of the square yard and be the radius of the circular yard. What is the expression that you want to optimize, in term of both and ?
(c) (4 points) Find an equation relating the variables and
(d) (7 points) Use your work from previous parts to design the yard(s) optimally. After your mathematical justification, sketch a picture of your finished set-up, with the dimensions labeled.
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Let R(x) = ( 2x^2 − 8x + 6 ) / (x^2 + 6x + 8)
a) Find all x such that R(x) is undefined. Find all x such that R(x) = 0. Evaluate R(0). Graph R(x) indicating all vertical and horizontal asymptotes and x and y intercepts.
b) Find the intervals on which ( x^2 + 2x + 12 ) / (x − 2) ≥ 2x + 5.
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Plot the limaçon defined by polar equation r=1+2sin(Ѳ) and find the area inside the limaçon but outside of the inner loop.
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2. Evaluate the following limits.
(a) lim x→1+
ln(x) /ln(x − 1)
(b) limx→2π
x sin(x) + x ^2 − 4π^2/x − 2π
(c) limx→0
sin^2 (3x)/x^2
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The function f(x)= x^−5 has a Taylor series at a=1 . Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.
In: Math
use a Taylor series to get the diverivative of f(x)=arctanx^3 and check for the interval of convergence. Is the interval of convergence for f' the same as the interval for f or different? Why?
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1. (a) Find the directional derivative of f at (2,1) in the direction from (2,1) to (6,-2). Show your work clearly.
(b) Find a unit vector u such that the directional derivative of f in the direction of u at (2,1) is zero. Show all work required to justify your answer.
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1. write a sixteen line about Taylor's theorem
2. write a haiku about exponential growth
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Find the Maclaurin series of the function
f(x)=(3x2)e^−4x
(f(x)=∑n=0∞cnxn)
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Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x , z^2 > on the region E bounded by the planes y + z = 2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:
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Estimate ∫^−3 −5 ?^2+5? ?? using midpoints for ?=4n=4 approximating rectangles.
∫^−3 −5 ?^2+5? ?? is approximately
Estimate ∫^3 2 2/? ?? using right endpoints for ?=3 approximating rectangles.
∫^3 2 2/? ?? is approximately
Consider the integral
∫102(4?^2+2?+6)??
(a) Find the approximation for this integral using left endpoints and ?=4
?4=
(b) Find the approximation for this same integral, using right endpoints and ?=4
?4=
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