Calculus

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

a.) find all x intercepts of f

4x^3+x^4=0

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