Document
... ( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting. The marks for each question has been indicated in the margin. ...
... ( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting. The marks for each question has been indicated in the margin. ...
graphical transformations
... mean it crosses the x-axis at that point. • The function may just touch the x-axis at that point: it is still a zero but does not cross. • Look at y = x2 + 6x + 9 • It factors (x + 3)(x + 3) and just touches at x = -3 • You can have more than one factor the same • Factor: y = x4 – 4x3 – 2x2 + 12x + ...
... mean it crosses the x-axis at that point. • The function may just touch the x-axis at that point: it is still a zero but does not cross. • Look at y = x2 + 6x + 9 • It factors (x + 3)(x + 3) and just touches at x = -3 • You can have more than one factor the same • Factor: y = x4 – 4x3 – 2x2 + 12x + ...
Mid-Term Exam - Stony Brook Mathematics
... Suppose not. Then the open sets Uj = X − Aj cover X. Since X is compact, only finitely many are needed, so X = U1 ∪ · · · ∪ Un for some n. But since Uj ⊂ Uj+1 , this implies that X = Un . However, X − Un = An 6= ∅, so this is a contradiction. 5. Let U ⊂ R be an open set. Prove that U is a countable ...
... Suppose not. Then the open sets Uj = X − Aj cover X. Since X is compact, only finitely many are needed, so X = U1 ∪ · · · ∪ Un for some n. But since Uj ⊂ Uj+1 , this implies that X = Un . However, X − Un = An 6= ∅, so this is a contradiction. 5. Let U ⊂ R be an open set. Prove that U is a countable ...