MATH 122B Study Guide - Midterm Guide: Round-Off Error, Tangent, Quotient Rule

Polynomials and Derivatives In this project you will explore an alternative polynomial. Th once (first, second,...). The drawback is that it takes a while to work through all the algebra and arrive at the desired results. You will be surprised by the simplicity of the method and will encounter topics related to this method in subsequent method for computing the derivatives of a e method is simple and relies on algebra alone. It also produces all derivatives at math classes. Definition: Polynomial A polynomial of a single real variable, x, is a function that can be written PCx)anxn +an-131-1a + ao where n is a nonnegative integer and ay are real numbers called the coefficients. If an 0, n is called the degree of the polynomial 1. State the Binomial Theorem. 2. Given the fourth degree polynomial P(x)42x3x5, Find P'(c), P"(c) P"(c), and P( (c) Given the fourth degree polynomial Px) +2x3 2+5, find P(u + c). Use the Binomial Theorem to expand P(u +c) and collect the terms to express P as a polynomial will real variable and coefficients dy(c) in terms of c 4. Find the relationship between the derivatives in part 2 and the coefficients di(c) in part3 Write a general formula for this relationship that would apply to any inth degree polynomial P. 5. For an nth degree polynomial P, write an expression for P(u + c) as we did in part 3; where u is the real variable and the coefficients are d,(c). Replace the u's with (x - c) and notice that you have an expression for P in terms of x again (P(x)- P(x-c+c) 6. Use the expression you found for an nm de gree polynomial P(x) in part 5 to show that your general formula from part 4 is true

1. Binomial Theorem: a t p" (e) = 12c2 + 12c-2 P" (c) 24c +12 p", (e) = 24

/ Math123FallTEast - MX Jalan - X e A Secure | https://bbcsulb.desirežlearn.com/dz//le/content/4041/4/viewcontent/4441994/View * @ : # Apps Pearson Sign In B Free Office 365(use fie D offices Hurs Fall-201 Y Question involving ar % Assignment (o -- Mat Q Fevica Cuestiors: CE Q Program Design &D CSI-117 midterm Flus Pctween the Panels: Math 123 - Exam 3 (Sample 1) 1. Determine whether the sequence converges or diverges. If it converges, find the limit. _ 3n - n (b) {1, -34-, ...} (e) {V3, V3/3, 3/3/3, ...} (a) on -2n + n 2. Determine whether the series converges or diverges. If it converges, find the sum. ( #1 () È 3. (a) Find the sum of the first 100 terms for the sequence I , and then find the sum s. Ln2 + 4n +3 n=1' (b) Express 0,1234 = 0.123423234... as a ratio of integers. 4. (a) The function f(x) = -1.1 Inx satisfies the hypothesis of the integral test (i.e., it is continuous, positive, and decreasing on [2, x)). Apply the integral test to determine if E converges or diverges. Sni. (b) Apply an appropriate test to determine whether converges or diverges. 5. Determine whether the series converges or diverges by applving an appropriate test. O P - + + 1 1 : 0 Type here to search рон в резема оо : » Ramna g« E 11:10 PM = - 11/21/2012

Al boxes with a squaro base, an open top,and a volume of 210 1 have a surface area given by s),2. 860 where x is the length of the sides of the base. Find the absolute minimum of the surface area function on the interval (0,00). What are the dimensions of the box with minimum surface area? Determine the derivative of the given function S(x). s'(x) The absolute minimum value of the surface area function is Round to three decimal places as needed.) The dimensions of the box with minimum surface area are a base of length ft and a height of ft (Round to three decimal places as needed.)

Suppose a tour guide has a bus that holds a maximum of 82 people. Assume his profit (in dollars) for taking n people on a city tour is P(n) -n(41-0.5n)-82. (Although P is defined only for positive integers, treat it as a continuous function.) a. How many people should the guide take on a tour to maximize the profit? b. Suppose the bus holds a maximum of 37 people. How many people should be taken on a tour to maximize the profit? a. Find the derivative of the given function P(n). P (n)- If the bus holds a maximum of 82 people, the guide should take people on a tour to maximize the profit. b. If the bus holds a maximum of 37 people, the guide should take people on a tour to maximize the profit.

Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p)-200-25p dD p says that at a price of p- 5, a quantity of D(6) -75 units of the commodily can be sold. The elasticty E do D of the demand gives the approximate percent change in the demand for every 1% change in the price. Complete parts (a) through (d). a. Compute the elasticity of the demand function D(p) 200-25p. b. If the price is S5 and increases by 6%, what is the approximate percent change in the demand? The percent change in demand is 96. (Round to one decimal place as needed.) c. Show that for the linear demand function DP = a-bp, where a and b are positive real numbers, the elasticity is a decreasing function for p 20 and p ta The elasticity is E(p)- ·