91Ó°ÊÓ

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { for } 0 \leq x<1 \\ 1 & \text { for } x=1 \\ 2 x-2 & \text { for } x>1\end{array}\right.\)

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{x^{2}+x-12}{x+4}, x \neq-4\)

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{\sqrt{9+x}-\sqrt{9}}{x}, x \neq 0\)

Revenue from Sales The owner of a photocopy store charges 7 cents per copy for the first 100 copies and 4 cents per copy for each copy exceeding \(100 .\) In addition, there is a setup fee of \(\$ 2.50\) for each photocopying job. (a) Determine \(R(x)\), the revenue from selling \(x\) copies. (b) If it costs the store owner 3 cents per copy, what is the profit from selling \(x\) copies? (Recall that profit is revenue minus cost.)

Use limits to compute the following derivatives. \(f^{\prime}(0)\), where \(f(x)=x^{2}+2 x+2\)

Write the equation of the line tangent to the graph of \(y=x^{3}\) at the point where \(x=-1\).

In Exercises \(33-36\), refer to a line of slope \(m\). If you begin at a point on the line and move \(h\) units in the \(x\) -direction, how many units must you move in the \(y\) -direction to return to the line? \(m=\frac{1}{3}, h=3\)

The third derivative of a function \(f(x)\) is the derivative of the second derivative \(f^{\prime \prime}(x)\) and is denoted by \(f^{\prime \prime \prime}(x) .\) Compute \(f^{\prime \prime \prime}(x)\) for the following functions: (a) \(f(x)=x^{5}-x^{4}+3 x\) (b) \(f(x)=4 x^{5 / 2}\)

In Exercises \(33-36\), refer to a line of slope \(m\). If you begin at a point on the line and move \(h\) units in the \(x\) -direction, how many units must you move in the \(y\) -direction to return to the line? \(m=2, h=\frac{1}{2}\)

The revenue from producing (and selling) \(x\) units of a product is given by \(R(x)=3 x-.01 x^{2}\) dollars. (a) Find the marginal revenue at a production level of 20 . (b) Find the production levels where the revenue is \(\$ 200\).