TBUS 500 Business Calculus Derivative and Marginal Costs Problems

Hi,

I need step by step solutions for all the problems in the attached document.

Thanks

TBUS 500
Derivatives Homework Problems
1) . (a) At what values of x does the graph of f in Fig. 1 have a horizontal tangent line?
(b) At what value(s) of x is the value of f the largest? smallest?
(c) Sketch the graph of m(x) = the slope of the line tangent to the graph of f at the point (x,y).
Figure 1
2) Match the graphs of the three functions in Fig. 2 with the graphs of their derivatives.
Figure 2
3) Find the derivatives of the following:
a) 𝑓(𝑥) = 4𝑥 3
2
b) 𝑓(𝑥) = 3𝑥 4
c) 𝑓(𝑡) = 3𝑡 2 − 2𝑡 − 5
4) A company’s total monthly sales (in millions of dollars) t months from now are given by:
𝑆(𝑡) = 0.5𝑡 2 + 3𝑡 − 5
a) Find S’(t)
b) Find S(4)
c) What does the value found in b) represent?
d) Find S’(4)
e) What does the value found in d) represent?
5) A company manufactures paper shredders. The total weekly cost (in dollars) of producing x shredders is given by:
𝐶(𝑥) = 14,500 + 35𝑥 − 0.02𝑥 2
a) Find the marginal cost function
b) Find the marginal cost at a production level of 400 shredders per week
c) Interpret the value found in b)
d) Find the exact costs of producing the 401st item
6) Stanley Bank offers a money market account that earns 1.85% compounded continuously:
a) If $10,000 is invested in this type of account, how much will it be worth in 3 years?
b) How long will it take for the account to be worth $15,000?
7) A manufacturer has determined that an employee with d days of production experience will be able to
produce approximately P(d) = 6 + 22( 1 – e–0.4d ) items per day.
(a) Approximately how many items will a beginning employee be able to produce each day?
(b) How many items will an experienced employee be able to produce each day?
(c) What is the marginal production rate of an employee with 3 days of experience? (What are the units of your
answer, and what does this answer mean?)
8) Find the derivatives of the following:
a) 𝑓(𝑥) = 4𝑒 𝑥 − 2𝑥 + 5
b) 𝑓(𝑥) = ln 2𝑥 2
c) 𝑓(𝑥) = (4 − 𝑥)5
9) An investment of $20,000 earns interest at an annual rate of 3.6% compounded continuously.
a) Find the instantaneous rate of change of the amount in the account after 3 years.
b) Find the instantaneous rate of change of the amount in the account when the amount is equal to $25,000.
10) Find the derivatives of the following:
a) 𝑓(𝑥) = 𝑒 6𝑥
b) 𝑓(𝑥) = 3𝑥 2 (5𝑥 3 + 7)
2𝑥
c) 𝑓(𝑥) = 𝑥+1