Chapter 7: Problem 10
Find the derivative of the function. $$ g(x)=3 x-1 $$
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You are managing a store and have been adjusting the price of an item. You have found that you make a profit of \(\$ 50\) when 10 units are sold, \(\$ 60\) when 12 units are sold, and \(\$ 65\) when 14 units are sold. (a) Fit these data to the model \(P=a x^{2}+b x+c\). (b) Use a graphing utility to graph \(P\). (c) Find the point on the graph at which the marginal profit is zero. Interpret this point in the context of the problem.
The ordering and transportation cost \(C\) per unit (in thousands of dollars) of the components used in manufacturing a product is given by $$ C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad 1 \leq x $$ where \(x\) is the order size (in hundreds). Find the rate of change of \(C\) with respect to \(x\) for each order size. What do these rates of change imply about increasing the size of an order? Of the given order sizes, which would you choose? Explain. (a) \(x=10\) (b) \(x=15\) (c) \(x=20\)
A population of bacteria is introduced into a culture. The number of bacteria \(P\) can be modeled by \(P=500\left(1+\frac{4 t}{50+t^{2}}\right)\) where \(t\) is the time (in hours). Find the rate of change of the population when \(t=2\).
Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=t^{2} \sqrt{t-2} $$
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