Spring 2005 MATH 172
Week in Review V
courtesy of David J. Manuel
Section 8.8, 8.9, 9.1
Section 8.8
1. State and derive the Trapezoid Rule.
1
(Ln + Rn ) = Tn (L, R, and T refer to Left rectangles, Right rectangles, and Trape2
zoids respectively).
2. Show that
Section 8.9
3. Use the precise definition of the limit (section 2.4 or 2.6) to precisely define the following:
Z
a)
∞
a
Z
b)
b
a
f (x) dx coverges to L
f (x) dx diverges to ∞ (where f is unbounded at x = a.)
4.
Theorem: If f, g continuous and 0 ≤ f (x) ≤ g(x) on [a, ∞) and
Z Prove part 2 of the Comparison
Z
∞
a
f (x) dx diverges, then
∞
a
g(x) dx diverges.
Z
5. Use the Comparison Theorem to show whether
1
∞
√
x
dx converges or diverges.
1 + x5
Section 9.1
6. Newton’s Law of Cooling states that the rate of change in the temperature of an object is proportional to the difference in temperature between the temperature of the object and its surroundings.
Given the initial temperature is y0 and the surrounding temperature is T , write and solve a differential equation to find the temperature of the object at any given time.
7. Given a family of curves, we can differentiate implicitly to find a differential equation y 0 = f (x, y)
for that family. The orthogonal trajectories of the family of curves is the family of curves such that
1
. Solving this differential equation gives us the orthogonal trajectories of the original
y0 = −
f (x, y)
family of curves. Find the orthogonal trajectories of the following families of curves:
a) xy = C
b) y = Cex