# Calculus

1. Let f(x) =4x^3+x^4

a.) find all x intercepts of f

4x^3+x^4=0

### Save your time - order a paper!

Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines

Order Paper Now

X^3(4+x) =0

X^3=0………………..i

4+x=0………………..ii

X^3=0

X=cube root of 0

X=0

4+x=0

X=-4

X intercepts=0,-4

X intercepts= (0, 0) (-4, 0)

b.)Find all intervals on which f is increasing and decreasing and find all points (x, y) at which relative extreme occur

4x^3+x^4=f(x)

Differentiate the function

12x^2+4x^3=0

4x^2(3+x) =0

(4x^2(3+x))/4=0/4

X^2(3+x) =0

3x^2+x^3=0

X^2(3+x) =0

X^2=0…………………i

3+x=0…………………..ii

X=0

X=-3

X= (0,-3)

Split (-∞, ∞) into separate intervals around the x values that make the derivative 0 or undefined.

Replace x with -1 in the expression

4(-1) ^3+ (-1) ^4=f (-1)

4*-1+1=f (-1)

-4-1=f (-1)

-5=f (-1)

=-5

When x=-1 the derivative is -5.this is a decreasing function

Ii.)Replace x with 4 in the expression

4(4) ^3+ (4) ^4=f (4)

256+256=f (4)

512=f (4)

=512

When x is 4 the derivative is 512.this is an increasing function

Increasing on: (∞, 0)

Decreasing on: (3, ∞)

c.)All points concave up, down and all points at inflection.

Concave up

F(x)>0

(4,512)

(2.8, 150.76)

Concave downward

F(x) <0

(-1,-5)

(-3,-189)

Inflection      f(x) =0

(0, 0)

d.) graph

1. Find all intervals on which f(x) =(x^2-x+16)/x-1 is increasing and decreasing and find all values of x at which relative extrema occur.

F(x) =(x^2-x+16)/x-1

(X-1)(X^2-x+16)

X^3-x^2+16x-x^2-x-16

X^3-2x^2+15x

Dy/dx=3x^2-4x+15

X=-b±sqrtb^2-4ac/2a

X=0,-1.33

FX>0= (1,-4)

FX<0= (-1,-34)

FX=0…… (0,-16)

3.) F(x) =x^4-12x^3+48x^2+2

Conc. up on= (2,114)

Conc.down= (-1,112)

Point of inf at x=0

4.) Find all vertical and horizontal asymptotes of g(x) = (x^2-4)/2x^3-x-10

X²-4

2x³-x-10

Vertical asymptotes                                       horizontal asymptotes

2x^3-x-10=0                                                      y=0 since the degree at the top is less than at the bottom

(2x+5)(X-2)=0                                                    x^2 is less than x^3

X=-5/2

X=2

5.) Find all absolute extrema of f(x) =100+9x+3x^2-x^3 on the interval (0, 4)

3x^2-x^3+9x+100=f(x)

3x^2-x^3+9x+100=0

Dy/dx=6x-3x^2+9

3(2x-x^2+3)

(X+3)(X-1)

X=-3

X=1

6.

1. Find the differential dy

a.) Y=ln(x^2+4)

Dy/dx=x^2/x+4

b.) y=x^2cos (5x)

y=x^2 –sin (5x)*5

Dy/dx=2-5 sin (5x)

=-3 sin (5x)

8.

V=4pie r^2

4*3.14*2^2

50.24cm ^3

803 tins

1. X^3-5x^2-8x+3

Dy/dx=3x^2-10x-8

First derivative…………………

(-b±sqrt b^2-4ac)/2a

X= (-1, 2.33)

Second derivative………………..

F(x)>0= (-1, 10), (-5, 45)

F(x) <0= (4,-45), (2.33,-32)

F(x) =0

10.)

\$20*x+3x^3=21600

3x^3+2x-21600=0

Dy/dx=9x^2+2=0

X^2=-2/9

X=0.22

3(0.22^3)+2(0.22)

0.007+0.44

0.447

21600/0.447

\$48322