Description
please see attachments for the practice problems and the class notes.
Make sure to show work to every step in the solution (look to the class notes)
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Math 172 Sections 8.2, 11.1-11.2 Practice Problems
Name:
Due: Wednesday, 10 June, 9pm (MDT)
30 total points, point values in boxes .
If possible, print this and submit it to gradescope. If not, please clearly write the problems and your
solutions on your own paper.
1. Parametric Equations.
2
(a) Express the curve described by the parametric equations x =
y = f (x) by eliminating the parameter.
√
t+2
and y = 6 − 2t in the form
4
3
(b) Find parametric equations x(t) and y(t) for the line segment starting at the point (1, 7) and ending
at (−4, 1). Include an appropriate domain for t.
2
(c) Find parametric equations x(t) and y(t) for the circle of radius 3 and center (2, −3).
8
2. A curve is defined by c(t) = cos2 (3t), 1 − 4t2 .
(a) Compute
dx
π
at t = .
dt
4
(c) Find the slope
dy
π
at t = .
dx
4
(b) Compute
dy
π
at t = .
dt
4
(d) Find the speed
ds
π
at t = .
dt
4
8
3. Consider the symmetric vertical plate bounded by the x-axis and the graphs of y = arccos x and
y = − arccos x. The plate is submerged in a fluid of density ρ with fluid level at y = π/2. See figure.
Find the fluid force on one side of the plate.
π/2
-1
fluid level
1
4. An important property of arc length is that it is independent of the parameterization. Below are two
parameterizations of the same line segment, show that each generate the same arc length.
3
(a) x = πt, y = 2t for t ∈ [0, 1]
4
(b) x = πet , y = 2et for t ∈ (−∞, 0]
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Explanation & Answer
Attached.
PRACTICE PROBLEMS
1. a) x = (t + 2)/4
t = 4x – 2
Substitute t = 4x – 2 into y = √(6 – 2t)
y = √(6 – 2[4x – 2])
y = √(6 – 8x + 4])
y = √(10 – 8x)
b)
(1,7)
(-4,1)
x starts at one and travels to the left for five units, hence the negative sign.
x = 1 + (-5) t
x = 1 - 5t, t ∈ [0,1]
y starts at seven and travels to the down for six units, hence the negative sign.
y = 7 + (-6) t
y = 7 - 6t, t ∈ [0,1]
When t = 0, x = 1 – 5(0)
When t = 0, y = 7 – 6(0)
x=1–0
y=7–0
x=1
y=7
When t = 1, x = 1 – 5(1)
When t = 0,...
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