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# Basic Calculus

### Mathematics - Calculus in Commerce

Production The output Q at a certain factory is related to inputs x and y by the equation: Q = x^3 + 2xy^2 + 2y^3 If the current levels of input are x = 10 and y = 20, use calculus to estimate the change in input y that should be made to offset an increase of 0.5 in input x so that output will be maintained at its cu

### Mathematics - Calculus in Commerce - Marginal Analysis

Marginal Analysis A manufacturer estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars, where C(x) = x^3 - 24x^2 + 350x + 338 a) At what level of production will the marginal cost C'(x) be minimized? b) At what level of production will the average cost A(x) = C(x)

### Mathematics - Calculus - Minimizing Cost.

An electronic firm uses 600 cases of transistors each year. Each case costs \$1000. The cost of storing one case for a year is 90 cents and the ordering fee is \$30 per shipment. How many cases should the firm order each time to keep total cost at minimum. (Assume that transistors are use at a constant rate throughout the year and

### Gross Domestic Product (GDP)

Gross Domestic Product (GDP) The gross domestic product of a certain country was N(t) = t^2 + 6t + 300 billion dollars t years after 1995. Use calculus to predict the percentage change in the GDP during the second quarter of 2003.

### Mathematics - Calculus, Functions and Intervals

Given the function f(x) = x^3 -3x-1 solve the following algebraically must show work algebraically Identify the intervals on which the following are true (1) Function is increasing (2) Slopes of tangent lines are positive (3) Function is concave up (4) Slopes of tangent lines are increasing ie becoming less n

### Mathematics - Calculus - Minimization

An open-topped cylindrical pot is to have volume 125in^3. What dimensions will minimize the total amount of material used in making this pot? Neglect the thickness of the material and possible wastage.

### Area Between Productivity Curves

After t hours on the job, one factory worker is producing Q'1(t) = 60 - 2(t - 1)^2 units per hour, while a second worker is producing Q'2(t) = 50 - 5t units per hour. a) If both arrive on the job at 8:00 AM how many more units will the first worker have produced by noon than the second worker? b) Interpret the answer in

### Mathematics - Calculus - Minimizing Cost

A surfboard manufacturing company has determined that the cost function c(x), in hundreds of dollars, for producing x hundreds of boards is c(x) = .2x^2 -.84x + 3.625. How many surfboards should the company make to minimize production costs?

### Mathematics - Application of Calculus in Commerece

A manufacturer estimates marginal revenue to be R'(q) = 100q^ -1/2 dollars per unit when the level of production is q units. The corresponding marginal cost has been found to be 0.4q dollars per unit. Suppose the manufacturer's profit is \$520 when the level of production is 16 units. What is the manufacturer's profit when the le

### Calculus

A long rectangular sheet of metal is to be made into a rain gutter by turning up two sides at right angles to the remaining center strip. The rectangular cross section of the gutter is to have area 18in^2. Find the minimum possible width of the sheet.

### Maximizing Area of a Fence

A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 640 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

### Function characteristics ... (Please see attached file)

For function ...(Please see attached file) a) identify zeroes of function b) what(if any) are vertical and horizontal asymptotes, in equation form c)first derivative: f'(x)= second derivative f"(x)= d) intervals where function is increasing and decreasing

### Instantaneous Rate of Change - For the Function

For the function: y(x) = x^2 - 4.1x + 1.7 find the instantaneous rate of change of the function at x = 2. Compare the instantaneous rate of change to the average rate of change on the interval [1.9, 2.0]. (Please see the attached file)

### Calculus to Determine Functions

Determine the given function is a solution of the differential equation: y = Ce^-t + 4 y' +y - 7 = 0 (Please see the attached file).

### Revenue Functions Modeled

The revenue for a given company as a function of time is modeled by the function: R(t) = -175.1t^2 + 193.5t + 295.3 Compute the rate of change of the revenue at any time for this model function and determine the predicted rate of change of revenue at t = 1 year and t = 5 years. (Please see the attached file).

### Mathematics - Calculus - Analyzing the Functions

Suppose we want to analyze the function: f(x) = 3x^4 - 8x^3 (a) Determine all points where the concavity of the function is zero. (b) Which of these points is inflection points? (Please see the attached file).

### Clarification of Mathematics - Calculus

Find the equation of a tangent line to the function at the indicated point: g(x) = e^-x^3 (-1, e^-1) (Please see the attached document)

### Mathematics - Calculus Sales Problem

Please help with the following problem. Provide a step by step calculation. Sales and Saturation Point : Integration using Separation of Variables - The rate of increase in sales ... (Please see the attached file) Sales: The rate of increase in sales S (in thousands of units) of a product is proportional to the current le

### Graphs of the functions and compute the area of this region

Sketch the region between the graphs of the functions and compute the area of this region: y = 9 - x^2, y = x^2 - 4

### Product and Minimum Sum

Determine two numbers that have a product of 128 and a minimum sum.

### Application of Derivatives

The prong of a tuning fork moves back and forth when it is set into vibration. The distance the prong moves between its extreme positions is 2.22 mm. If the frequency of the tuning fork is 441.0 Hz, what are the maximum velocity and the maximum acceleration of the prong? Assume SHM. ______m/s ______m/s^2^

### Optimization ...

Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs \$1/ft^2; the material for the base costs \$2/ft^2. FInd the dimensions of the box of greatest possible volume if you are allowed to spend \$144 for the material to make it.

### Curve Sketching ...

Sketch the graph of y = x^4/4 - 2x^2 + 1. i) determine and show on your graph all relative minima, maxima, and points of inflection. ii) find intervals where the graph is increasing, decreasing, concave up, and concave down. iii) find the absolute maximum and absolute minimum values of y on the interval [-1,3].

### Riemann Sums : Evaluating an Integral ...

How do you use Riemann sums to evaluate an integral? I don't have an example written, but could provide one if necessary.

### Rate of Growth of an Investment

Investment: The rate of growth of an investment is proportional to the amount in the investment at any time t. That is, dA/dt = kA The initial investment is \$1000 and after 10 years the balance is \$3320.12. The general solution is: A = Ce^kt What is the particular solution?

### Description of Mathematics - Calculus

Use integration to find the general solution of the differential equation: dy/dx = 1 / (1 + x)

### Mathematics - Calculus - Maximum Value..

8. An entrepreneur buys an apartment building with 40 units. The previous owner charged \$240 per month for a single apartment and on average rented 32 apartments at that price. The entrepreneur discovers that for every \$20 he raises the price another apartment stands vacant. A. Let x represent the number of \$20 price increases

### Mathematics - Calculus - Functions

C(x)=2x+56 R(x) = 20x-x2 a)Find the marginal cost function ?what does this function predict? b)Find the profit function. c)Find the marginal profit function ?what does this function predict? d)What are the breakeven points? e)Find the value of x, where the graph R(x) has a horizontal tangent.

### Mathematics - Calculus - Maximum Value

A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If 85 m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

### Mathematics - Calculus - Rate measures

A plane flying at an altitude of 25,000 ft. has a defective airspeed indicator. To determine her speed, the pilot sights a fixed point on the ground. At the moment when the angle of depression (From the horziontal) of her line of sight is 65 degrees, she notes that this angle is increasing at 1.5 degrees per second. What is the