Download Math Exercise: Length of Parametric Curves, Cycloids, and Area of Surfaces and more Exams Advanced Calculus in PDF only on Docsity!
Exercise #2 (practice problems for math 253)
- If f ′(x) is continuous, we know that the length of graph y = f (x) with a ≤ x ≤ b is given by L =
∫ (^) b a
1 + (f ′(x))^2 dx. If the graph is given by parametric notations, that is, for α ≤ t ≤ β,
consider the graph determined by
x = x(t), y = y(t)
. How to find the length of that part of the graph?
a) In the original form of arc lenth, it is given as
∫ (^) b a
1 + ( d dyx )^2 dx. Now we have parametric
notation instead, but note that d dyx =
dy dt dx dt
y′(t) x′(t)
. Also note that d dxt = x′(t), how can we get the
length of the graph with α ≤ t ≤ β in terms of x(t) and y(t)?
b) If we have x′(t) > 0, what will the formula of the length be?
c) The cycloid, the parametric equation is
x = r(t − sin t), y = r(1 − cos t). For 0^ ≤^ t^ ≤^2 π, find the length of the cycloid (that is, find the length of one period for the cycloid).
- For
x = a cos t, y = b sin t , what is the slope of the tangent line at t = π 4?
Hint: It is exactly d dyx
t= π 4 , which is the same as
dy dt dx dt
t= π 4.
- For
x = a cos t, y = b sin t with 0 ≤ t ≤ π, if we rotate the graph around x-axis, what is the area of
the surface we will get?
Hint: You can find the standard function notation y = f (x) first, then use the formula to find the area of the surface arising from rotating a curve around x-axis. Or, you can try this approach: We know that the area of the surface arising from rotating a curve arounf x-axis is given by
A =
∫ (^) b
a
2 π|y| dL =
∫ (^) b
a
2 π|y|
1 + (f ′(x))^2 dx =
∫ (^) b
a
2 π|y|
dy dx )^2 dx.
Note that d dyx =
dy dt dx dt
and dx = x′(t)dt, we will get
A =
∫ (^) β
α
2 π|y|
dy dt dx dt
)^2 · x′(t)dt
- For
x = a cos t, y = b sin t with 0 ≤ t ≤ 2 π, find the area surrounded by this curve.
Hint: Consider the part of the curve above x-axis. The area surrouned by that part of curve and the x-axis is
∫ (^) a −a y^ dx. Using the parametric notation, we have
A =
∫ (^) a
−a
y dx =
π
y(t)x′(t) dt.
n.
- If xn → 0 +, is it true that xnn → 0? Is it true that n
xn → 0?
- Find limx→∞(
x^2 + x − x). (Hint:
x^2 + x − x =
√ x^2 +x−x 1 , rationalize the numerator.)
- ** For the series
∑^ ∞
n=
(−1)n^
1 + 12 + · · · + (^) n^1 n , is it convergent or divergent? (Hint: 1 + 12 + · · · + 1 n = 1^ ·^ 1 +^
1 2 ·^ 1 +^ · · ·^ +^
1 n ·^ 1, using the idea of integral test, can we find a upper bound and lower bound for 1 + 12 + · · · + (^) n^1 ?)
∑^ ∞
n=
(−1)n^
n
n
, is it convergent or divergent?
Find the length of the graph of y = x^2 for 0 ≤ x ≤ 2.
For the series
∑^ ∞
n=
3 n^ + en 2 · en^ + n^3 , is it convergent or divergent?
- For the power series
∑^ ∞
n=
xn n(n + 1) , find the radius of convergence R. For x = R and x = −R,
is the series convergent? Why?
- For the power series
∑^ ∞
n=
xn 5 n^ + 7n^
, find the radius of convergence R.
Find the Maclaurin series for f (x) = sin x.
- Find the Maclaurin series for f (x) = arcsin x.
For the series
∑^ ∞
n=
n(n + 1)
, is it convergent or divergent? Why?
- True or false: If
∑^ ∞
n=
an is convergent, then Σ(−1)nan is also convergent.
- True or false: If
∑^ ∞
n=
an is divergent, then an 9 0.
- For the series
∑^ ∞
n=
n ·
n + 1
, is it convergent or divergent? Why?
- For the series
∑^ ∞
n=
(−1)n^ sin(n^2 ) n^2
, is it convergent or divergent? Why?
