Math 241 - Spring 2024 Banwarth Name as it appears in Canvas: Lab H OSUID: Print and complete this lab (or complete it on your tablet). Then scan and upload it to Gradescope in the corresponding Lab Assignment. Conceptual questions 1. A company knows that x units of a product will sell at a price of p dollars per unit. The function p(x) outputs the per unit price in dollars for an input of the number of units sold, x. The solution to which of the following equations could find the number of units, x, that will maximize the revenue for this company? ○ xp'(x) + p(x) = 0 ○ p'(x) = 0 x-p'(x)-p(x) x² = 0 ○ p" (x) = 0 Ox p(x) = 0 2. The total cost of producing x units for a company can be modeled by the function C(x), where C(x) is in dollars. The solution to which of the following equations could find the number of units, x, that will minimize the average cost of production? Ox C'(x) + C(x) = 0 • ○ C'(x) = 0 x-C'(x)-C(x) x² = 0 OC" (x) = 0 C(x) = 0 x Math 241 Lab H Spring 2024 Math skills questions 1.) Use the chain rule to differentiate the following functions. 3 a.) f(x)=√√x²+1 f'(x) = b.) f(x)=(x3 - 2)4 f'(x) = c.) f(x) = (1-x)² f'(x) = d.) f(x) = √√(x³ + 1)² − 4 f'(x) = 1 e.) f(x) = x²+1 (Note: You could use the quotient rule, but for practice you should rewrite it as (x²+1)¹ and use the chain rule.) -1 f'(x) = x f.) f(x) = = x²+1 . (Note: You could use the quotient rule, but for practice you should rewrite it as x* (x² + 1)−¹ and use the product rule in combination with the chain rule.) f'(x) = Math 241 Lab H Spring 2024 2.) Derivatives of exponential functions Find the derivatives of the following exponential functions by rewriting them so they have the form Cekx for some constants C and function k, then using the constant multiple and chain rules to differentiate. Remember that bx: eln b*x for any b > 0. d d = Example: (32x) = (9x) = 1 4 (eln 9+x) = ln 9 * e¹n 9** = ln 9 * 3* dx d dx dx Note: there are many ways to rewrite exponentials. I could have written 32x as (3*)² and differentiated using a different set of steps, but even if the answer looked different, it would have been the same. a.) — (5³x) Answer: d b.) (ex²+1) dx Answer: c.) (53x²+1) dx Answer: d.) [(53x+1)²] Answer: Math 241 Lab H Spring 2024 Applied questions Exponential functions are a very important set of functions because they define how most natural systems grow and/or shrink. For example, biological populations, nuclear reactions, investment assets, friend networks. Any function whose instantaneous rate of change is directly proportional to the function itself is an exponential function (growth or decay). Put another way, any system that changes according to the size of the system is exponential. 1.) Explain how the equation f'(x) = k × f(x) is a mathematical description of the previous two sentences. Answer: 2.) Banks make money through taking deposits, then lending those deposits to borrowers and charging them an interest rate. In order to get enough deposits, banks in turn give depositors a (smaller) interest rate. The difference between the lending interest rate and the deposit interest rate constitutes the bank's revenue. a) Suppose you deposit $5000 in an account that pays you 2% annual interest. Each year the bank adds the interest to your account, which therefore grows over time. Write down the function that computes the account balance in your account over time (let the input variable be t). B(t) = = b) What will your balance be after 8 years if you don't withdraw any of the funds? Answer: Math 241 Lab H Spring 2024 c) To make money, the bank will take (most) of the $5000 and loan it out, trusting that you don't want all of it back right away. Suppose that in year one your bank lends $4,500 of your initial deposit to Grey's Anatomy, an amusement park whose only ride is a giant statue of Elliot Grey with a slide inside his digestive system. A bank's reserve ratio is the proportion of the deposit that is not lent out by the bank. Determine the reserve ratio for this bank given your initial deposit and how much they lend to Grey's Anatomy. Answer: d) Grey's Anatomy will use that money to upgrade the surface of the slide. But they don't have to pay the contractor right away. So they deposit the $4,500 in their bank, (where they also earn an interest rate, but don't worry about that now). Grey's bank turns around and lends the same ratio of Grey's deposit to Nowhere Man Enterprises, a company that makes maps. How much money is Nowhere Man Enterprises lent by the bank? Answer: e) As you can imagine, the trend continues: each subsequent borrower initially parks their loan in a bank, and that bank turns around and lends out the deposit minus the reserve ratio to another business. Write down the formula for L(n), which is the amount of money loaned to the nth business (Grey's Anatomy is the first business). Assume the initial deposit is $5000 and the reserve ratio for every bank is what you computed in part c). L(n) = f) What kind of function is L(n)? ○ Linear ○ Exponential growth ○ Exponential decay ○ Square root ○ Quadratic ○ None of these Math 241 Lab H Spring 2024