- In each of the following, assume the relation defines y as a function of x for values
of x and y of interest and find y
^{′}.- xy
^{2}+ sin= x^{3}+ 1 - y
^{3}+ xcos= x^{4} - y cos= tancos+ 2
^{6}= x^{3}y + 3- + cos= 7
- sin= 3 x
- y
^{3}sin+ y^{2}x^{2}= 2^{x2 }y + ln - y
^{2}sinx + log_{3}= y^{2}+ 11 - sin+ sec= e
^{x+y}+ y2^{y}+ 2 - sin+ y
^{3}= 16 - cos+ ln= x
^{2}y + 3

- xy
- In each of the following, assume the relation defines y as a function of x for values of x
and y of interest. Use the chain rule to show y satisfies the given differential
equation.
- x
^{2}y + siny = 7,y^{′}+ 2xy = 0. - x
^{2}y^{3}+ sin= 5 , 2xy^{3}+y^{′}= 0. - y
^{2}sin+ xy = 6,

- x
- Show that if D⊆ U ⊆ D, and if f and g are both one to one, then f ∘ g is also one to one.
- The number e is that number such that lne = 1. Prove e
^{x}= exp. - Find a formula for for y = b
^{x}. Prove your formula. - Let y = x
^{x}for x ∈ (0,∞). Find y^{′}. - The logarithm test states the following. Suppose a
_{k}≠0 for large k and that p = lim_{k→∞}exists. If p > 1, then ∑_{k=1}^{∞}a_{k}converges absolutely. If p < 1, then the series, ∑_{k=1}^{∞}a_{k}does not converge absolutely. Prove this theorem. - Suppose f= f+ fand f is continuous at 0. Find all solutions to this functional equation which are continuous at x = 0. Now find all solutions which are bounded near 0. Next if you want an even more interesting version of this, find all solutions whose graphs are not dense in the plane. (A set S is dense in the plane if for every∈ ℝ × ℝ and r > 0, there exists∈ S such that
This is called the Cauchy equation.

- Suppose f= ffand f is continuous and not identically zero. Find all solutions to this functional equation. Hint: First show the functional equation requires f > 0.
- Suppose f= f+ ffor x,y > 0. Suppose also f is continuous. Find all solutions to this functional equation.
- Using the Cauchy condensation test, determine the convergence of ∑
_{k=2}^{∞}. Now determine the convergence of ∑_{k=2}^{∞}. - Find the values of p for which the following series converges and the values of p for
which it diverges.
- For p a positive number, determine the convergence of
for various values of p.

- Determine whether the following series converge absolutely, conditionally, or not at all
and give reasons for your answers.
- ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n}sin - ∑
_{n=1}^{∞}^{n}tan - ∑
_{n=1}^{∞}^{n}cos - ∑
_{n=1}^{∞}^{n}sin

- ∑
- De Moivre’s theorem says
for n a positive integer. Prove this formula by induction. Does this formula continue to hold for all integers, n, even negative integers? Explain.

- Using De Moivre’s theorem, show that if z ∈ ℂ then z has n distinct n
^{th}roots. Hint: Letting z = x + iy,and argue

is a point on the unit circle. Hence z =. Thenis an n

^{th}root if and only if^{n}= z. Show this happens exactly when=and α =for k = 0,1,,n. - Using De Moivre’s theorem from Problem 15, derive a formula for sinand one for cos.
- Suppose ∑
_{n=0}^{∞}a_{n}^{n}is a power series with radius of convergence r. Show the series converge uniformly on any intervalwhere⊆. - Find the disc of convergence of the series ∑
for various values of p. Hint: Use Dirichlet’s test.
- Show
for all x ∈ ℝ where e is the number such that lne = 1. Thus

Show e is irrational. Hint: If e = p∕q for p,q positive integers, then argue

is an integer. However, you can also show

- Let a ≥ 1. Show that for all x > 0, you have the inequality

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