Calculus Assignment

Calculus Assignment

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. This course is a challenging technical course intended for those majoring in economics or business.

It is really helpful for the students to learn the subsequent courses such as intermediate microeconomic theory, econometrics, statistics, etc. Students will extend their experience with functions as they study the fundamental concepts of calculus: limiting behaviors, difference quotients and the derivative, Riemann sums and the definite integral, anti-derivatives and indefinite integrals, and the Fundamental Theorem of Calculus. It covers limits, continuity, differentiation with applications of finding relative (or absolute) maximum and minimum values, sketching curve, for the functions of one variable such as polynomial, rational, exponential, and logarithmic functions, with an emphasis on applications to problems from business and economics.

On the other hand, it covers integration with their applications to find area under a graph and solve differential equations; to find total function (total cost function) from the marginal function (marginal cost function), find consumer and supplier surplus in economics; to find expected value, standard variance in probability theory. Finally, it introduces functions of several variables, partial derivatives, unconstrained and constrained maximum and minimum for functions of several variables with their applications in business and economics. Below is the solution of the given course work.

Answer Number # 01

Find the absolute extrema, if they exist, as well as all values of x where they occur for the function.

F(x) = , on the closed interval [-2, 2]

The first step is to find df(x)/dx. For this, we are using the quotient rule.

df(x)/dx =

Now, set df(x)/dx to 0.= 0

= 0

Now find f(0) and f(3)

f(-2) :

=

= 20

f(+2) :

Answer: Absolute max 20 at x= -2; Absolute min -60 at x= +2.

Answer Number # 02

A closed box with square base is to have a volume of 288 in3. The material for the base costs $1 per square inch whereas the sides and top cost only $0.50 per square inch. Find the dimensions of the box which will lead to the minimum total cost.

The given volume is;

V = 288 = wLh

h = 288/(wL)

The cost is given by;

C = w*L*1 + w*L*0.5 + 0.5(2*w*h + 2L*h)The base must be quadratic, because a square has a maximum area at minimum perimeter, and a cube holds a maximum volume at minimum area. So an elongated cube is the second best structure for maximum volume. And in addition, the four sides have the same prize.This makes the equation to;

C =

C =

C =

Therefore,

L = 5.76 cm

W = 5.76 cm

H = 288/ L^2 = 8.68 cm

Answer: The side of the base should be 5.76 cm and the height should be 8.68 cm.

Answer Number # 03

The cost of manufacturing x capacitors per day is C(x) = 0.5x + 120e -0.001x + 23,000 dollars. How fast is the cost changing ...