Answer to Question #179129 in Calculus for Michael Faustino

Question

1. Suppose the marginal cost for producing x units of a commodity is given by dC/dx= 32 − 0.04x. If it costs $50 to make 1 unit, find the total cost of making 200 units.
2. Find the function f whose tangent has slope 3x^2 + 6x − 2 and whose graph passes through the point (0,6).
3. The methods of differentiation and antidifferentiation are widely used in estimating various quantities in real-world. In economics, experts use tools like marginal cost and marginal revenue as its basic necessities. Calculus is used in this field in calculating the change in demand for a product, calculating change in its cost price, and estimating the rate of change in its revenue with an increase in its selling price. What do you think will happen if the people hired by the government to do these tasks lack expertise in the said skill?

Answer 1

1.     Suppose the marginal cost for producing x units of a commodity is given by 

$\frac {dc}{dx} = 32-0.04x ......... (i )$ Then, the total cost C (x) expression is found by integrating (i) with respect to x

$C (x)=\int(32-0.04x)dx = 32x-0.02x^2+k ........ (ii)$

Where k is a constant corresponding to the fixed cost. We are further given that C (1)=50 Thus 

$50=32 (1)-0.02 (1)^2+k$

Giving k=18.02 Therefore, substituting this value of k in the expression for C, we have

$C (x)= 32x-0.02x^2+18.02$

Which gives us 

$C (200)=32 (200)-0.02 (200)^2+18.02 = 5618.02$

=$5618.02

2. For the function f (x), we are given that

$f'(x)=3x^2+6x-2 .......... (i)$

Now, integrating (i) with respect to x., we have

$f (x)=x^3+3x^2-2x+c ......... (ii)$

Where c is a constant of integration. We too are given that the graph of this function passes the point (0,6). This means that f (0)=6

Then 6=f (0)=c

The function so described is thus

$f (x)=x^3+3x^2-2x+6$

3.     The methods of differentiation and anti-differentiation are widely used in estimating various quantities in real-world. In economics, experts use tools like marginal cost and marginal revenue as its basic necessities. Calculus is used in this field in calculating the change in demand for a product, calculating change in its cost price, and estimating the rate of change in its revenue with an increase in its selling price. What do you think will happen if the people hired by the government to do these tasks lack expertise in the said skill?

Various effects of hiring employees that lack expertise will be directly imposed to the government. The following are some of these effects:

a)      Poor models lead to unrealistic government policies. These policies on implementation end up in lowering the quality of performance at the work places by the employees and therefore, this damages the reputation of the government.

b)    Budget makers need expertise in calculus to be able to make sound estimates. Employees who lack calculus skills will put forth estimates that are sensible. Organizations that may want to work with any government will lose trust in such a case they are to receive such no sensible estimates. Government expenditures too would be prudent in case it is guided by policies founded on calculus. This will not be the case in cases where a government has hired employees short of these skills and the ultimate price will be huge government loses.