MTH282
Question
Every nonempty set of real numbers of that has an upper bound, also has a supremum in R. This is the _____
Answer
The Completeness
Question
Using method of mathematical induction prove this \(3 + 11 + \cdots + (8n-5) =\) ________ for all \(n\in N\)
Answer
\(n(4-n)\)
Question
\(\lim_{n\rightarrow 0} sin\frac{1}{n}\) =
Answer
does not exist
Question
What is the first derivative of the function \(f(x)=x^{n}\)
Answer
\(nx^{n-1}\)
Question
Find the derivatives of the function \(f(x)=x^{2}sin\frac{1}{x}\) when \(x\neq 0\), and f(x)=0 when x=0
Answer
\(0\)
Question
Suppose f(x), g(x) exist, \(g^{1}(x)\neq 0\) and f(x)=g(x)=0. So, \(\lim_{t\rightarrow x}\frac{f(t)}{g(t)}=\)_______________
Answer
\(\frac{f^{1}(x)}{g^{1}(x)}\)
Question
Let g be the function defined by \(g(x)=e^{2x+1}\) for all real x. Then \(\lim_{x\rightarrow 0}\frac{g(g(x)-g(e))}{x}\)
Answer
\(4e^{2e+2}\)
Question
Let S, T and U be nonempty sets and let \(f :S\rightarrow T\) and \(g: T\rightarrow U\) be functions such that the function \(gof:S\rightarrow U\) is one to one (injective). Which of the following must be true
Answer
f is one to one
Question
If f is a continuously differentiable real valued function defined on the open interval (1,-4) such that f(3) = 5 and \(f^{1}(x) \geq -1\) for x, what is the greatest possible value of f(0)
Answer
8
Question
Evaluate \(\lim_{x\rightarrow 0}\frac{cos(3x)^{-1}}{x^2}\)
Answer
-9/2
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