Math 1320 Lab Review
Name:
1. A cardboard box without a lid is to have a volume of 32, 000 cm3 . Find the dimensions
of the box that minimize the amount of cardboard used.
2. Let R be the rectangle which has the largest area of all possible rectangles with a fixed
perimeter p. Using Lagrange multipliers, prove that R is a square. (This is a specific
example of a very general phenomenon.)
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3. Consider the simple model of animal growth, which assumes that the animal is a cube
of side length s. The growth of the animal is governed by the differential equation
ds
= αs2 − βs3 ,
dt
where α and β are constants related to the rate of food intake and the cost of body
maintenance, respectively. Suppose α = 0.05, β = 0.2.
(a) What is the order of this differential equation?
(b) Sketch the vector field of this differential equation.
(c) Is the differential equation separable? If so, solve the differential equation with
initial condition s(0) = 1.
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4. Does the sequence {1, −1/2, 1/3, −1/4, . . .} converge? Why or why not? Does the
sequence {1, 1 − 1/2, 1 − 1/2 + 1/3, 1 − 1/2 + 1/3 − 1/4, . . .} converge? Why or why not?
In your justifications, cite relevant theorems and show how they apply.
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5. For which values of x do the following series converge?
∞
X
cos(x)
(a)
n2 + 1
n=0
(b)
(c)
5000
X
x
n
n=1
∞ X
n=0
(d)
(e)
∞
X
n=0
∞
X
x−1
4
2n
(1 + n)xn
n−1 ln(n)−x
n=2
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6. Find a power series for the following function:
f (x) =
d 4 −x2 3x e
dx
Does the power series converge for x = 10000?
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7. Find a Taylor series for f (x) = x3 about a = 1. Find an upper bound on the error from
approximating f (x) by T1 (x) in 0.9 ≤ x ≤ 1.1.
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8. Given vectors a, b, and c, and scalar d, determine whether the following are vectors or
scalars, or if the operation is not defined.
(a) a · b
(b) a × b
(c) |a − b|
(d) d + a
(e) da
(f) a/b
(g) a × b × c
(h) a · (b × c)
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9. Sketch the vectors a = (−1, −1/2, 0) and b = (−3, 1, 0). Does a × b point into or out of
the paper? What does the magnitude of the cross product represent? Sketch a + b and
a − b. Compute a · b and |2a − b|. Compute projb a and (a − projb a) · projb a. How do
you interpret this result?
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10. Sketch the vector function r(t) = (cos(t), sin(t), arctan(t)).
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11. Write the equation of the plane that goes through (1, 4, 2), (2, 6, 6), and (0, −4, 3) in
linear, vector, and scalar form.
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p
12. Sketch somep
level curves of f (x, y) = cos 1 + x2 + 2y 2 . Hint: Find the level curves
of g(x, y) = 1 + x2 + 2y 2 first. Because cosine is periodic, every level curve of f (x, y)
will correspond to infinitely many disjoint level curves of g(x, y).
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13. Use what you know about polar coordinates and the chain rule to compute
f (r, θ) = 2r2 θ.
Your answer should be in terms of r and θ.
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∂f
∂x
and
∂f
:
∂y
14. Classify the extrema of the function f (x, y) = x4 + y 4 − 2x2 .
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15. Find the tangent plane to the graph of the function in the previous problem when
(x, y) = (1, 1).
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