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Answers to Exam 1 - Mathematical Analysis for Business I | MATH 130, Exams of Mathematics

Material Type: Exam; Class: Mathematical Analysis for Business I; Subject: Mathematics; University: Ohio State University - Main Campus; Term: Winter 2007;

Typology: Exams

Pre 2010

Uploaded on 07/23/2009

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MATH 130 Answers to Test 1 (Form B) Autumn 2007
1. Solve the given equation.
(i) x+ 6
x2โˆ’3xโˆ’1
x=x
xโˆ’3.
x(xโˆ’3) x+ 6
x(xโˆ’3) โˆ’x(xโˆ’3) 1
x=x(xโˆ’3) x
xโˆ’3
x+ 2 โˆ’(xโˆ’3) = x(x); x+ 6 โˆ’x+ 3 = x2;
9 = x2;x= 3 or x=โˆ’3.
Check: If x= 3, x+ 6
x2โˆ’3xโˆ’1
x=9
0โˆ’1
3is undefined.
If x=โˆ’3, x+ 6
x2โˆ’3xโˆ’1
x=โˆ’3+6
(โˆ’3)2โˆ’3(โˆ’3) โˆ’1
โˆ’3= 0.5;
x
xโˆ’3=โˆ’3
โˆ’3โˆ’3= 0.5. Answer:x=โˆ’3
(ii) Solve โˆšy+ 6 = y+ 4.
y+ 6 = (y+ 4)(y+ 4) = y2+ 4y+ 4y+ 16; 0 = y2+ 8y+ 10 = (y+ 2)(y+ 5). So y=โˆ’2
or y=โˆ’5.
If y=โˆ’2, โˆšy+ 6 = โˆšโˆ’2 + 6 = 2 and y+ 4 = โˆ’2 + 4 = 2.
If y=โˆ’5, โˆšy+ 6 = โˆšโˆ’5 + 6 = 1 but y+ 4 = โˆ’5 + 4 = โˆ’1. Answer:y=โˆ’2
2. Solve the inequality 2(xโˆ’1) โˆ’3(x+ 1) โ‰ค5. Give your answer using interval notation.
2xโˆ’2โˆ’3xโˆ’3โ‰ค5; โˆ’xโ‰ค10; xโ‰ฅ โˆ’10. Answer: [โˆ’10,โˆž)
(over) Typeset using AMS-T
E
X.
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MATH 130 Answers to Test 1 (Form B) Autumn 2007

  1. Solve the given equation.

(i)

x + 6 x^2 โˆ’ 3 x

x

x x โˆ’ 3

x(x โˆ’ 3) x + 6 x(x โˆ’ 3) โˆ’ x(x โˆ’ 3)

x = x(x โˆ’ 3) x x โˆ’ 3

x + 2 โˆ’ (x โˆ’ 3) = x(x); x + 6 โˆ’ x + 3 = x^2 ; 9 = x^2 ; x = 3 or x = โˆ’3.

Check : If x = 3, x + 6 x^2 โˆ’ 3 x

x

is undefined.

If x = โˆ’3, x + 6 x^2 โˆ’ 3 x

x

(โˆ’3)^2 โˆ’ 3(โˆ’3)

x x โˆ’ 3

= 0.5. Answer: x = โˆ’ 3

(ii) Solve

y + 6 = y + 4. y + 6 = (y + 4)(y + 4) = y^2 + 4y + 4y + 16; 0 = y^2 + 8y + 10 = (y + 2)(y + 5). So y = โˆ’ 2 or y = โˆ’5. If y = โˆ’2,

y + 6 =

โˆ’2 + 6 = 2 and y + 4 = โˆ’2 + 4 = 2. If y = โˆ’5,

y + 6 =

โˆ’5 + 6 = 1 but y + 4 = โˆ’5 + 4 = โˆ’1. Answer: y = โˆ’ 2

  1. Solve the inequality 2(x โˆ’ 1) โˆ’ 3(x + 1) โ‰ค 5. Give your answer using interval notation.

2 x โˆ’ 2 โˆ’ 3 x โˆ’ 3 โ‰ค 5; โˆ’x โ‰ค 10; x โ‰ฅ โˆ’10. Answer: [โˆ’ 10 , โˆž)

(over) Typeset using AMS-TEX.

  1. Let f (x) = x^2 โˆ’ 4 x.

(i) Find f (โˆ’2). Simplify your answer as much as possible. (โˆ’2)^2 โˆ’ 4(โˆ’2) Answer: 12

(ii) Find f (x + h). Simplify your answer as much as possible. (x + h)^2 โˆ’ 4(x + h) = (x + h)(x + h) โˆ’ 4 x โˆ’ 4 h = x^2 + xh + hx + h^2 โˆ’ 4 x + 4h Answer: x^2 + 2xh + h^2 โˆ’ 4 x โˆ’ 4 h

(iii) Find f (x + h) โˆ’ f (x) h

. Simplify your answer as much as possible. x^2 + 2xh + h^2 โˆ’ 4 x โˆ’ 4 h โˆ’ x^2 + 4x h

2 xh + h^2 โˆ’ 4 h h Answer: 2x + h โˆ’ 4

  1. Find the domain of the given function.

(i) h(x) =

3 โˆ’ 7 x.

3 โˆ’ 7 x โ‰ฅ 0; 3/ 7 โ‰ฅ x. Answer: (โˆ’โˆž, 3 /7]

(ii) g(x) = x + 1 x^2 โˆ’ 4

. Answer: All numbers except ยฑ2.

  1. Let F (x) = x^2 โˆ’ 2 x and G(x) = x โˆ’ 3.

(i) Find (F G)(x). Simplify your answer as much as possible.

(x^2 โˆ’ 2 x)(x โˆ’ 3) = x^3 โˆ’ 3 x^2 โˆ’ 2 x^2 + 6x Answer: x^3 โˆ’ 5 x^2 + 6x

(ii) Find (F โ—ฆ G)(x). Simplify your answer as much as possible.

F (G(x)) = F (x โˆ’ 3) = (x โˆ’ 3)^2 โˆ’ 2(x โˆ’ 3) = x^2 โˆ’ 6 x + 9 โˆ’ 2 x + 6 Answer: x^2 โˆ’ 8 x + 15

  1. Find functions f (x) and g(x), neither of which is the identity function, such that (f โ—ฆ g)(x) =

x + 2.

f (x):

x g(x): x + 2

(go to the next page)

  1. A lumber company owns a wooded area that is in the shape of a rectangle that measures 2 miles by 3 miles. They are planning to clear a uniform strip along the outer edges of the forest, so that 3 square miles of forest remain.

3 miles

2 miles

x

x

x x

(i) Write down a quadratic equation for the width of the sttrip. Let the width of the strip be x. Then the area of the forest that remains uncut is (2 โˆ’ 2 x)(3 โˆ’ 2 x). So we want (2 โˆ’ 2 x)(3 โˆ’ 2 x) = 3; 6 โˆ’ 4 x โˆ’ 6 x + 4x^2 = 3; Answer: 4x^2 โˆ’ 10 x + 3 = 0

(ii) Solve the quadratic equation to find the width of the strip. Give your answer to the nearest 0.01 mile.

x =

(โˆ’10)^2 โˆ’ 4(4)(3)

โ‰ˆ 2 .15 or 0. 35

The width of the strip cannot be more than 2/2 = 1 mile. Answer: 0.35 mile

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