Topic 2: Logic and useful definitions Flashcards
What is the difference between being necessary and sufficient (2)
-If being in A implies being in B (A is a subset of B), A is a sufficient condition for B (if it is in A it must also be in B)
-If being in A implies being in B, B is a necessary condition for A (has to be in B to be in A)
What is the contrapositive theorem (1)
-If A implies B, not being in B implies not being in A
What is deductive vs inductive reasoning (2)
-Deductive reasoning is reasoning based on consistent rules of logic
-Inductive reasoning attempts to infer general conclusions from observations
How can we define a function being continuous (2)
-f(x) is continuous at x=a if the lim x->a f(x) = f(a)
-This requires f(a) to be defines, lim x->a to exist, and the limit to = f(a)
what is the ε - δ definition (3)
-The function f(x) has a limit A as x tend to a, if for every number ε > 0, there exists a number δ > 0 such that |(f(x) - A)| < ε for every x with 0 < |(x-a)|< δ
-ε = error in the measurement at the value of the limit, δ = distance to the limit
-What this means is as you approach a limit, if you place a y value error measurement, there will always be an x value approaching the limit where you’ll be closer to the limit then the error measurement
What is a necessary and sufficient condition for the limit at x = a to exist (1)
-A necessary and sufficient condition for the limit at x = a to exist is the right and left limit to exist and be equal
What is L’Hopital’s rule (1)
-If f and g are differentiable functions over interval (a, b), and suppose f(x) and g(x) both tend to 0 as x -> a, the limit as x->a of f(x)/g(x) = the limit as x -> a of f’(x)/g’(x)
What is the relationship between being differentiable and continuous at x = a (1)
-If f is differentiable at x = a, f is continuous (diff suff for continuity, not necessary due to a curve with a vertical tangent being undifferentiable)
