Calculus
Answers doc
- 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
- 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.
- 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
- 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
