91Ó°ÊÓ

Find each derivative. $$\frac{d}{d x}(-2 \sqrt[3]{x^{5}})$$

Draw a graph that is differentiable and has horizontal tangent lines at \(x=0, x=2,\) and \(x=4.\)

Differentiate each function. $$f(x)=\left(\frac{2 x}{x^{2}+1}\right)^{3}$$

Is the function given by \(f(x)=3 x-2\) continuous at \(x=5 ?\) Why or why not?

The function \(p(t)=\frac{2000 t}{4 t+75}\) models the population \(p\) of deer in an area after \(t\) months. a) Find \(p^{\prime}(10), p^{\prime}(50),\) and \(p^{\prime}(100)\) b) Find \(p^{\prime \prime}(10), p^{\prime \prime}(50),\) and \(p^{\prime \prime}(100)\) c) Interpret the meaning of your answers to parts (a) and (b). What is happening to this population of deer in the long term?

Use the Chain Rule to differentiate each function. You may need to apply the rule more than once. $$f(x)=\left(2 x^{5}+(4 x-5)^{2}\right)^{6}$$

The Candy Factory sells candy by the pound, charging \(\$ 1.50\) per pound for quantities up to and including 20 pounds. Above 20 pounds, the Candy Factory charges \(\$ 1.25\) per pound for the entire quantity, plus a quantity surcharge \(k .\) If \(x\) represents the number of pounds, the price function is \(p(x)=\left\\{\begin{array}{ll}1.50 x, & \text { for } x \leq 20 \\ 1.25 x+k, & \text { for } x>20\end{array}\right.\) a) Find \(k\) such that the price function \(p\) is continuous at \(x=20.\) b) Explain why it is preferable to have continuity at \(x=20.\)

A lab technician controls the temperature \(T\) inside a kiln. From an initial temperature of 0 degrees Celsius \(\left(^{\circ} \mathrm{C}\right),\) he allows the kiln to increase by \(2^{\circ} \mathrm{C}\) per minute for the next 60 min. After the 60th minute, he allows the kiln to cool at the rate of \(3^{\circ} \mathrm{C}\) per minute. The temperature function \(T\) is defined by \(T(t)=\left\\{\begin{array}{ll}2 t, & \text { for } t \leq 60 \\ k-3 t, & \text { for } t>60\end{array}\right.\) a) Find \(k\) such that \(T\) is continuous at \(t=60\) b) Explain why \(T\) must be continuous at \(t=60\) min.

Compound interest. If \$ 1000\( is invested at interest rate i, compounded annually, in 3 yr it will grow to an amount A given by (see Section R.1) \)A=\$ 1000(1+i)^{3}. a) Find the rate of change, \(d A / d i\) B) Interpret the meaning of dA/di.

The cost of sending a large envelope via U.S. first-class mail is \(\$ 0.88\) for the first ounce and \(\$ 0.17\) for each additional ounce (or fraction thereof). (Source: www.usps.com.) If \(x\) represents the weight of a large envelope, in ounces, then \(p(x)\) is the cost of mailing it, where $$\begin{array}{l} p(x)=\$ 0.88, \quad \text { if } \quad 0< x \leq 1, \\ p(x)=\$ 1.05, \quad \text { if } \quad 1 < x \leq 2, \\ p(x)=\$ 1.22, \quad \text { if } \quad 2 < x \leq 3, \end{array}$$ and so on, up through 13 ounces. The graph of \(p\) is shown below Using the graph of the postage function, find each of the following limits, if it exists. $$\lim _{x \rightarrow 2^{-}} p(x), \lim _{x \rightarrow 2^{+}} p(x), \lim _{x \rightarrow 2} p(x)$$