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Chapter 7: Problem 18
Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6 ;(-2,6) $$
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Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{3}, u=3 x^{2}-2 $$
Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\sqrt{4 x^{2}-7} $$
The cost of producing \(x\) units of a product is given by \(C=x^{3}-15 x^{2}+87 x-73, \quad 4 \leq x \leq 9\) (a) Use a graphing utility to graph the marginal cost function and the average cost function, \(C / x\), in the same viewing window. (b) Find the point of intersection of the graphs of \(d C / d x\) and \(C / x\). Does this point have any significance?
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(t)=\frac{t+2}{t^{2}+5 t+6} $$
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) \(x=20\)
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