Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
3rd Edition
ISBN: 9780840064189
Author: Frederick R. Adler
Publisher: Cengage Learning
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Chapter 2.9, Problem 22E
a.
To determine
To find derivative of the function
F(x)=(1+2x)3
By expanding the binomial
a.
Expert Solution
Answer to Problem 22E
F′(x)=24x2+24x+6
Explanation of Solution
Given:
F(x)=(1+2x)3
Concept Used:
Addition/subtraction rule ofdifferentiation formula- Power rule of differentiation formula
- (a+b)3=a3+b3+3a2b+3ab2
ddx(f(x)±g(x))=ddxf(x)±ddxg(x)
ddxxn=nxn−1
Calculation:
To find derivative of the function
F(x)=(1+2x)3
First expand the binomial as
F(x)=(1+2x)3F(x)=13+(2x)3+3(2x)2(1)+3×12×2xF(x)=1+8x3+12x2+6x
Now, differentiate it, as
ddx[F(x)]=ddx(1+8x3+12x2+6x)F′(x)=ddx1+ddx(8x3)+ddx(12x2)+ddx(6x)F′(x)=0+8ddx(x3)+12ddxx2+6ddxxF′(x)=8(3x3−1)+12(2x2−1)+6(1)F′(x)=24x2+24x+6
Hence,
F′(x)=24x2+24x+6
b.
To determine
To find derivative of the function
F(x)=(1+2x)3
By using chain rule.
b.
Expert Solution
Answer to Problem 22E
F′(x)=24x2+24x+6
Explanation of Solution
Given:
F(x)=(1+2x)3
Concept Used:
- Chain rule of differentiation is: ddxf[g(x)]=ddgf(g)dgdx , where g is a differentiable function at x and f is a differentiable function at g(x)
- Power rule of differentiation formula
ddxxn=nxn−1
Calculation:
To find derivative of the function
F(x)=(1+2x)3
Use chain rule of differentiation that is,
ddxf[g(x)]=ddgf(g)dgdx
Where, f(x)=x3; g(x)=(1+2x)
Thus, differentiation of given function can be calculated as
ddx[F(x)]=ddx(1+2x)3F′(x)=3(1+2x)3−1ddx(1+2x)F′(x)=3(1+2x)2(ddx1+2ddxx)F′(x)=3(1+(2x)2+4x)(0+2(1))F′(x)=(3+12x2+12x)2F′(x)=24x2+24x+6
Hence,
F′(x)=24x2+24x+6
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Chapter 2 Solutions
Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
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- Use the method of undetermined coefficients to solve the given nonhomogeneous system. x-()*+(5) = 1 3 3 1 X+ t +3 -1 -2t 1 x(t) = º1 1 e +021 e +arrow_forwardFind the general solution of the given system. 6 -(-1)x x' = -6 11 x(t) = x(t) = e5t)*[(c1 + c2(t− 1/6))(c1 + c2t)] Your answer cannoarrow_forward(c) Describe the distribution plan and show the total distribution cost. Optimal Solution Amount Cost $ 2000 Southern-Hamilton 200 Southern-Butler $ Southern-Clermont 300 4500 Northwest-Hamilton 200 $2400 Northwest-Butler 200 $3000 Northwest-Clermont $ Total Cost ક (d) Recent residential and industrial growth in Butler County has the potential for increasing demand by 100 units. (i) Create an updated distribution plan assuming Southern Gas becomes the preferred supplier. Distribution Plan with Southern Gas Amount Southern-Hamilton $ Cost × Southern-Butler x $ Southern-Clermont 300 $ 4500 Northwest-Hamilton 64 x Northwest-Butler $ × Northwest-Clermont 0 $0 Total Cost $ (ii) Create an updated distribution plan assuming Northwest Gas becomes the preferred supplier. Distribution Plan with Northwest Gas Southern-Hamilton Southern-Butler 0 Southern-Clermont Northwest-Hamilton Northwest-Butler Northwest-Clermont Total Cost Amount × x x +7 $0 Cost × $ × $ × +4 $ -/+ $ × ×arrow_forward
- The distribution system for the Herman Company consists of three plants, two warehouses, and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows. Warehouse Plant Capacity 1 2 1 4 7 450 2 8 5 600 3 5 6 380 Customer demand and shipping costs per unit (in $) from each warehouse to each customer are as follows. Customer Warehouse 1 2 3 1 6 4 8 2 3 6 7 7 Demand 300 300 300 400 (a) Develop a network representation of this problem. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen This answer has not been graded yet. (b) Formulate a linear programming model of the problem. (Let Plant 1 be node 1, Plant 2 be node 2, Plant 3 be node 3, Warehouse 1 be node 4, Warehouse 2 be node 5, Customer 1 be node 6, Customer 2 be node 7, Customer 3 be node 8, and Customer 4 be node 9. Express your answers in the form x;;, where x,; represents the number of units shipped from node i to node j.) Min 4x14+8x24+5x34+7x15 +5x25…arrow_forwardA linear programming computer package is needed. Hanson Inn is a 96-room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. A large profesional organization has scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least 50% of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two-night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are…arrow_forward25.2. Find the Laurent series for the function 1/[z(z-1)] in the follow- ing domains: (a). 0<|z|< 1, (b). 1<|z, (c). 0arrow_forward
- 25.5. Find the Laurent series for the function 1/[(z - 1)(-2)(z - 3)] in the following domains: (a). 0 3. شهریarrow_forward25.1. Expand each of the following functions f(z) in a Laurent series on the indicated domain: (a). z² - 2z+5 (2-2)(z² + 1)' (c). Log za 2 b (z - موجود 11, 29, where b>a> 1 are real, |z| > b.arrow_forward25.3. Find the Laurent series for the function z/[(22 + 1)(z² + 4)] in the following domains (a). 02.arrow_forward
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