769 words - 4 pages

Name: Bekenov Kuandyk

Assignment 1

1. Prove that fx=2x+5 is one-to-one function. (10 points)

Let make an assumption tha fx1=fx2t and then prove that x1=x2.

fx1=fx2=2x1+5=2x2+5→x1=x2 ,

Then fx=2x+5 is one-to-one function

2. B

B

A

A

Let f: A→B, as given below. Is f a one-to-one function? Please explain why or why not. (10 points)

f

f

5

5

5

5

1

1

3

3

1

1

6

6

2

2

2

2

4

4

6

6

7

7

4

4

8

8

3

3

F is not one-to-one function, because f(1)=1 and f(2)=1.

3. The modulo function (a mod n or a modulo n) maps every positive integer number to the remainder of the division of a/n. For example, the expression 22 mod 5 would evaluate to 2 since 22 divided by 5 is 4 with a remainder of 2. The expression 10 mod 5 would resolve to 0 since 10 is divisible by 5 and there is not a remainder.

a. If n is fixed ...view middle of the document...

Find lim2x99+x+1,000,000x100+2,000,000 as x→0 (6 points)

limx→02x99+x+1000000x100+2000000=0+0+10000000+2000000=0.5

8. Prove that fx=x2+7 is a continuous function. (8 points)

Let prove that fx+∆x-fx→0 when ∆x→0 then:

x+∆x2+7-x2+7=x2+2x∆x+∆x2+7-x2-7-=∆x2x+∆x=0, it means fx= x2+7 is a continuous function.

9. Find derivatives of the following functions using differentiation rules. (Do not use the definition of a derivative!)

c. fx=2x+5 (3 points)

f'x=2x'+5'=2

d. fx=2x5+x3-6x+100 (3 points)

f'x=10x4+3x2-6

e. fx=ex+2(4x-1) (3 points)

f'x=ex+24x-1+4ex+2=ex+2(4x+3)

f. fx=7ex (3 points)

f'x=7ex

g. fx=x2(x-1) (3 points)

f'x=2xx-1+ x2=x(3x-2)

h. fx=3x2+7 (3 points)

f'x=6x

i. fx=ex(2x+5) (3 points)

f'x=ex2x+5+2ex=ex(2x+7)

j. fx=ex+2+4x-1 (3 points)

f'x=ex+2+4

10. Find the derivative of the following function at :fx=2x3+x2+9 (8 points)

f'-1=6x2+2x=6-2=4

11. Find the derivative of the following function fx=2x2-x using the definition of a derivative. (Hint: though you can use the rules of differentiation to check your answer, you must use the definition of a derivative to solve this problem in order to receive any credit for your response) (10 points)

lim∆x→0fx+∆x-f(x)∆x=lim∆x→02(x+∆x)2-x+∆x-(2x2-x)∆x=lim∆x→02x2+4x∆x+2∆x2-x-∆x-2x2+x∆x=lim∆x→04x∆x+2∆x2-∆x∆x=lim∆x→04x+2∆x-1=4x-1

Complementary problems (worth up to 3 extra credit points each)

1. Let fx=2x and gx=ex, where. Find and as well as the domain and range of these functions.

f ogx=fgx=2ex,

gofx=gfx=e2x

domain of these functions: x∈-∞,∞

rangeof these functions: any real power of any positive number is always positive, whether the exponent is positive, negative or 0.So the range is “all positive numbers”,

or in interval notation (0, ∞)

2. Find the derivative of by using the definition of the derivative

lim∆x→0(x+∆x)-1-x-1∆x=lim∆x→0x-x-∆xx(x+∆x)∆x=lim∆x→0-∆xx∆x(x+∆x)=lim∆x→0-1x2+x∆x=-1x2

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