Week 6

Question 1

Induction

Answer 1

if for all integers n there is a function P(n), we can prove P(n) is true if
1. P(1) is true
2 if P(n) is true, then P(n+1) is also true

Question 2

How to induction

Answer 2

Prove P(19 by LFS = RHS
use P(n) and add n+1, then try to get the same result using P(n+1)

Question 3

What if P(1) is not valid

Answer 3

if P(k) is true, induction can be used to prove P is true for n in N such that n is greater or equal to k

Question 4

Implicit diffrentiaton

Answer 4

application of chain rule. Derentiate y in terms of x y ading a dy/dx factor. Solve for dy/dx

Question 5

Second derivative implicit

Answer 5

same shit, derivaive of dy/dx is second derivative. sub in dy/dx when done

Question 6

logarithmic differentiation

Answer 6

take natural logs of both sides, use properties of logs to make equations easier, differentiate implicitly.

Question 7

Taylor series

Answer 7

sum to infity of (thing). expansion of the function f expressed about a in terms of only x. Mcclarin series when a=0

Question 8

nth degree taylor polynomial of f about a

Answer 8

partialsum of taylor expansion up to n. is a polynomial of degree n

Question 9

multiplication and division of series

Answer 9

if the taylor series for a function is known, similar functions can be multiplied by that. Te taylor expansion of a composite function is the product of the taylor functions of the composite functions.

Question 10

Strong induction

Answer 10

if p(1), p(2), p(k) are true, then p(k+1) is true. similar to normal induction for cases whre you have to use more than one prior examples.