Math Problems Solutions And Answers Essay

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Question:

Find the first derivative of the following functions and calculate their slope in points????= 1 and ???? = -1:

1. ???? = 4 + 2

3????? ????3

dy/dx= -4/3x-4/3 -6x-4

When x=1

-4/3*1-4/3 -6*1-4

-4/3-6= -7.3

When x= -1

-4/3*-1-4/3 -6*-1-4

4/3+6= 7.3

1. ???? = ?7????+5

4????2+1

-7(4X2+1)-8X(7X+5)

(4X2+1)2

-28X2-7+56X2-40X

(4X2+1)(4X2+1)

-7(4X2+1)-8(7X2-5X)

(4X2+1)(4X2+1)

When x=1

-28*1-7+56-40 =-19/25 =-0.76

25

When x=-1

-28*1-7+56+40 =61/25 =2.44

25

1. ???? = ln(3????2 + 5)

y=lnU

dy/dU= 1/U

dU/dx = 6x

dy/dx = 1/U*(6x)

= 6x

(3????^2 + 5)

When x=1

6*1/(3*1^2 + 5)=0.75

When x=-1

6*-1/(3*-1^2 + 5)=-3

Question : 2

1. Find the first derivative of the following functions:

1. ????(????) = ln(????????????)

(????) = ln(????????????)

F(x) = 1/lnx *1/x

=1/xlnx

1. ????(????) = 3???? ? ????????????3 + ????????

(????) = 3???? ? ????????????3 + ???? ????

f'(x)= xln3(3????)-3/x- ????????

1. Find the second derivative of the following functions:

1. ????(????) = ????????+ ????2

f'(x) = ex +2x

f"(x) = ex+2

1. ????(????) = ln(???? ? 1)

(????) = ln(???? ? 1)

f'(x) = 1*1/(x-1) =1/(x-1)

f"(x) = [0*(x-1)-1(1)]/(x-1)2

= -1/(x-1)2

Question : 3

Use the rule of the chain to find the first derivative of the functions:

1. ???? = 2???? ? 1

???? = 2???? ? 1

dz/dy = 2

dy/dx = 1+e^x

dz/dx = 2(1+e^x)

= 2(1+e^x)

1. ???? = ????????????????

???? = ???? ????????????

dz/dy = y/y= 1

dy/dx = 1+e^x

dz/dx = 1(1+e^x)

= 1+ e^x

1. ???? = 1 ? 4????2

???? = 1 ? 4????2

dz/dy = -8y

dy/dx = 1+ex

dz/dx = -8y(1+ ex)

= -8 (???? + ????x)(1+ ex)

=-8 (x+xex+ex+e2x )

with regard to...…= ????2 + ????????????4

2

f'(x) = x+4/x

=x^2 + 4

f'(x) = x+4/x

=x2 + 4

No point of inflection

[Suggestion. You do not have to make their graphic representations. Justify at what intervals the cavities are above or below and find the turning points].

Question: 5

Solve the following integrals:

1. ? ( ????2???? + 1 + 2 )????????

????? ????

=e2x+2lnx+x1/2

2

1. ? ????ln(????) ????????

lnx =U

du/dx =1/x

dx= xdu

V=x^2/x. Udv=(UV)'-VDU

? ???? ln(????) ????????

=x^2lnx- ? ????^2/2*1/x ????????

=x2lnx/2 x2/4 +C

1. ? 6????????2+1????????????

????????2+1

=2x derivative of power of (e)

(6 ????????2+1)

2x

=3????????2+1+C

1. ? 2????????????

????2?4

=lnx2-4+C

Question : 6

Calculate the integrals:

a. ?-1 (????3 ? 5???? + 1)6 (3????2 ? 5)????????

-2

?-1 (x3 ? 5x + 1)6 (3x2 ? 5)dx

-2

U=(x3-5x+1)6

du= 6(x3-5x+1)5(3x2-5)

dx

?-1 […

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