P4

Question 1

Proof: there’s no greatest odd integer

Answer 1

Assumption: There is a greatest odd integer, n

consider a number, n+2, greater than n, also an odd integer, which contradicts to assumption.

Question 2

Proof: There are infinitely many prime numbers.

Answer 2

Assumption: There is a finite number of prime numbers, which are P1, P2…Pn
Consider a number p=P1·P2·…Pn+1
So P must be either prime or has a prime factor which isn’t in the list of all possible prime numbers, contradicts with assumption

Question 3

Proof: if n^2 is even, then n must be even

Answer 3

Assumption: there exists a number n which is odd, but n^2 is even.
Let n=2k+1, n^2=(2k+1)^2=2(2k^2+2k)+1
n^2 is odd, which contradicts with assumption

Question 4

Proof: √(2) is an irrational number

Answer 4

Assumption: √(2) is a rational number
Let √(2)=p/q p, q are integers with no common factors
2=p^2/q^2, p^2=2q^2, so p^2 is even, then p must be even
Let p=2k, 2=(2k)^2/q^2, 4k^2=2q^2, q^2=2k^2
so q^2 is even, q must be even

Question 5

Binomial expansion for (1+x)^n?

Answer 5

(1+x)^n=1+nx+n(n-1)x^2/2!+n(n-1)(n-2)x^3/3!

Question 6

Differentiation of Parametric equation?

Answer 6

dy/dx=dy/dt÷dx/dt

Question 7

What’s an implicit equation?

Answer 7

Hard to turn into the form of y=f(x)

Question 8

Integration by parts?

Answer 8

∫(uv’)dx=uv-∫(vu’)dx

Question 9

The order of determining “u” when integrating by parts?

Answer 9

1. inverse trigonometric function: arcsinx, arccosx…
2. logarithms: lnx
3. power of x: x^n (includes 1=x^0)
4. trigonometric function: sinx, cosx…
5. exponential function: a^x, e^x…

Question 10

integration by substitution steps?

Answer 10

let u=…
express the integration with du
determine limits

Question 11

area=?

volume of revolution of x-axis=?

Answer 11

area=∫ydx

volume=Σπy^2γx=π∫y^2dx