Name: Last __________________ First ____________________________Section # _________

Signature Assignment Math 1316 Spring 2022

Due Date: Wednesday April 27, 2022 at 11:59 pm

[There will be a 20 point penalty assessed for turning in late]

The due date for this assignment is 11:59 pm on Wednesday April 27, 2022 at 11:59 pm. It is to be

uploaded as an online submission on Canvas. On Canvas go to Modules and then click on the

Signature Assignment Link located there. This link will let you download a copy of the assignment.

Then when you have finished the assignment click on submit assignment and a link will open up that

will let you upload your completed assignment. It is preferred that you upload it as a scanned pdf file

but if this is not possible you can upload it as a jpeg file by using your phone.. Use whatever method

works best for you – but make sure to have the completed assignment uploaded to Canvas no later

than 11:59 pm on April 27, 2022 .

1) Use the definition for the derivative 𝑓 ′(𝑥) = limℎ→𝑜 𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ to find the derivative of the

function f(x) = 3x 2 + 10x – 5. You must show all of your work. You can take the derivative the quick

way to check your answer – but you must show how to compute the derivative using the above

formula to get credit for this problem. (10 points)

𝑓 ′(𝑥) = _______________________________________

2) Assume that the cost equation in dollars of producing x units of a product is given by the equation

C (𝒙)=𝟐𝟓𝟎𝟎+𝟏𝟎𝒙 and that the monthly demand equation for the product is given by the equation

𝑝 = 30 − 𝑥

1000 where x is the number of units demanded per month when the price charged is

p dollars. Use the above information to compute the monthly revenue equation for the product.

Then find the monthly profit equation for the above product and use it to compute the monthly

marginal profit function for this product. Finally use this to determine the profit and marginal profit

associated with a monthly production level of 8000 units. All work must be shown! (16 points)

Revenue Equation = R(x) = ____________________________

Profit Equation = P(x) = _______________________________

Profit (8000) = ______________________________________

MP(8000) = _________________________________________

3) Given 𝑦=(x+2)(2𝑥 2 +3)

3 find the equation of the tangent line to this function when x = 1.

First find the point on this function and the slope of the tangent line to this function when

x = 1. Next use these to find the equation of the tangent line to this function when x = 1.

Finally, put this equation in slope intercept form. All work must be shown!! (15 points)

Point on function when x = 1 is (1, _____)

Slope of tangent line when x = 1 is _____________ Equation of tangent line in slope intercept form is: _______________________________________

4) A farmer needs to enclose by fences a rectangular field containing 392,000 square feet. One side of the field lies along a river and one side lies along a road. The river is perpendicular ( at right

angles) to the road, He needs a more expensive fence on the sides next to the road and the river,

with a cheaper fence being used on the remaining 2 sides. The fence costs $20 per foot along the

river, $15 per foot along the road, and $5 per foot on the remaining 2 sides to be fenced. Find the

dimensions he should use for the field to fulfill the above criteria and minimize the cost of fencing the

field. Also calculate the minimum cost possible for constructing a field that meets the above criteria?

Include a diagram of the field as part of your work. This diagram will help you determine the

correct cost function for constructing the fence. All work must be shown in order to get credit for

the problem!!!. (20 points)

The side next to the road must equal _____________ feet

The side next to the river must equal _____________ feet

The minimum cost of the fence is ___________________

5) Use integration by parts to evaluate the following integral :

∫ [3x 2 ln(x

3 )]dx. Identify the values that need to be used for u, du, dv, and v as well as

the final value for the integral. All work must be shown. (20 Points)

U = ___________________

du = ___________________

dv = ___________________

v = ____________________

∫ [3x 2 ln(x

3 )]dx = _____________________________

6a) Explain in your own words the meaning of the terms Consumer and Producer Surplus.

(4 Points)

6b) Given the demand equation for a product is p=D(𝑞)=200−0.3𝑞 2

and the supply function is

𝑝=S(𝑞)=0.2𝑞 2

find the equilibrium point for this product and use it to compute the value of the

Consumer and Producer Surplus associated with this product. All work must be shown. Round

answers off to two decimal places.

(15 points)

Equilibrium point _________________________

Consumer’s Surplus _______________________

Producer’s Surplus _______________________