Problem 1
A function \(y=f(x)\) is discontinuous at \(x=x_{0}\) when any of the three requirements for continuity is violated at \(x=x_{0} .\) Construct three graphs to illustrate the violation of each of those requirements.
Problem 1
Solve the following inequalities: (a) \(3 x-1 < 7 x+2\) (b) \(2 x+5 < x-4\) (c) \(5 x+1 < x+3\) \((d) 2 x-1 < 6 x+5\)
Problem 1
Given the function \(q=\left(v^{2}+v-56\right) /(v-7),(v \neq 7),\) find the left-side limit and the right-side limit of \(q\) as \(v\) approaches 7 . Can we conclude from these answers that q has a limit as \(v\) approaches \(7 ?\)
Problem 1
Given the function \(y=4 x^{2}+9\) (a) Find the difference quotient as a function of \(x\) and \(\Delta x\). (Use \(x\) in lieu of \(x_{0}\).) (b) Find the derivative \(d y / d x\) (c) Find \(f^{\prime}(3)\) and \(f^{\prime}(4)\)
Problem 1
Find the limits of the function \(q=7-9 v+v^{2}:\) (a) As \(v \rightarrow 0\) (b) As \(v \rightarrow 3\) (c) As \(v \rightarrow-1\)
Problem 2
Find the limits of \(q=(v+2)(v-3)\) (a) As \(v \rightarrow-1\) (b) As \(v \rightarrow 0\) (c) As \(v \rightarrow 5\)
Problem 2
If \(8 x-3 < 0\) and \(8 x > 0,\) express these in a continued inequality and find its solution.
Problem 2
Taking the set of all finite real numbers as the domain of the function \(q=g(y)=v^{2}-\) \(5 v-2\) (a) Find the limit of \(q\) as \(v\) tends to \(N\) (a finite real number). (b) Check whether this timit is equal to \(g(N)\) (c) Check whether the function is continuous at \(N\) and continuous in its domain.
Problem 2
Given the function \(y=5 x^{2}-4 x\) : (a) Find the difference quotient as a function of \(x\) and \(\Delta x\). (b) Find the derivative \(d y / d x\) (c) Find \(f^{\prime}(2)\) and \(f^{\prime}(3)\)
Problem 3
Solve the following: (a) \(|x+1| < 6\) (b) \(|4-3 x| < 2\) \((c)|2 x+3| \leq 5\)
