The cost of producing and selling x toasters each week is, given by c=1785+3x+0.002x^2 dollars. Find dc/dx and interpret its meaning.
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Looks like some serious studying ahead.
To find dc/dx, we need to differentiate the cost function with respect to x. Differentiating the given cost function c=1785+3x+0.002x^2 with respect to x involves applying the power rule and the constant rule of differentiation.
First, let's break down the term 0.002x^2:
The derivative of 0.002x^2 with respect to x is 0.002 * 2x = 0.004x. However, the derivative of the constant term 1785 is zero since it does not involve x. Finally, the derivative of the term 3x with respect to x is simply 3.
Adding these derivatives together, we find:
dc/dx = 0 + 3 + 0.004x = 3 + 0.004x.
Now, let's interpret the meaning of dc/dx:
The derivative dc/dx represents the rate of change of the cost function with respect to the number of toasters produced and sold each week. In this case, dc/dx = 3 + 0.004x.
The constant term, 3, represents the fixed cost component of producing and selling toasters each week, which does not vary with the number of toasters produced. This includes expenses such as rent, utilities, and overhead costs.
The term 0.004x represents the variable cost component, which increases proportionally with the number of toasters produced. Specifically, for each additional toaster produced, the variable cost increases by 0.004 dollars. This term accounts for direct costs such as labor, raw materials, and other expenses directly related to producing toasters.
Therefore, dc/dx = 3 + 0.004x represents the total cost of producing and selling each additional toaster beyond the fixed cost component.
To find the derivative of c with respect to x, we need to apply the power rule for derivatives.
Given that c = 1785 + 3x + 0.002x^2, we can differentiate each term separately.
The derivative of a constant is zero, so the derivative of 1785 is 0.
The derivative of 3x is simply 3.
To differentiate 0.002x^2, we bring down the exponent as a coefficient and reduce the exponent by 1, resulting in 0.004x.
Adding all the derivatives together, we have:
dc/dx = 3 + 0.004x
Interpreting the meaning of dc/dx:
The derivative dc/dx represents the rate of change of the cost with respect to the number of toasters produced and sold each week. In this case, the derivative is given by 3 + 0.004x.
The constant term 3 represents the fixed cost associated with producing and selling the toasters. It remains constant regardless of the number of toasters produced.
The term 0.004x represents the variable cost associated with producing and selling toasters. It indicates how the cost varies as more toasters are produced. The coefficient 0.004 represents the rate at which the cost increases per unit of x (toasters). Therefore, for each additional toaster produced and sold, the cost increases by 0.004 dollars.