Math Problems Solutions and Answers

Question:

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

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

-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

When x=-1

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

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

y=lnU

dy/dU= 1/U

dU/dx = 6x

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

= 6x

When x=1

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

f'(x)= xln3(3????)-3/x- ????–????

1. Find the second derivative of the following functions:

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:

dz/dy = 2

dy/dx = 1+e^x

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

= 2(1+e^x)

dz/dy = y/y= 1

dy/dx = 1+e^x

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

= 1+ e^x

dz/dy = -8y

dy/dx = 1+ex

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

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

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

with regard to ????, where ???? = ???? + ????????. Simplify the derivative as much as possible.

Question: 4

Find the spaces where the following functions are curved or concave and determine the turning points of their graphs.

a.(????) = 3????5 ? 40????3 + 3???? ? 20

Y=3x5-40x3+3x-20

x-intercept

X3(3x2-40)= 0

X3=0 then x=0

(3x2-40)= 0 then 3x2=40

X=3.65

y-intercept

When x=0 then y=-20

Maxima and minima

dy /dx=15x4-120x2+3

=5x4-40x2+1

X2(5x2-40)=0

X2=0 then x=0

5x2=40 then x=81/2 x=2.32 There is a maxima or minima at

X=0 y=-20

X=2.32 y=-310

Second order condition

y”=20x3-80x

When x=0 y”=0

X=2.32 y”>0 local minima (2.32, -310)

Points of inflection

y”= 20x3-80x then 20x(x2-4)

20x(x2-4)=0

X=0 x2=4 then x=2 If x=2 y”=-238

(2,-238)

b.(????) = ????2 + ????????????4

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:

=e2x+2lnx+x1/2

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

=2x derivative of power of (e)

2x

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

=lnx2-4+C

Question : 6

Calculate the integrals:

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

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

U=(x3-5x+1)6

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

dx

?-1 [ U (3x2 ? 5)du ]