find derivatives, eqns of tangent lines, slopes, etc.

[willkilla]

We started getting into derivatives and I was out a few days. Anyone care to give me some direction on these?

1. Let f(x) = -x^3 - 2x^2 + x + 1 and g(x) = ln(x + 1) + 1
a. Find the equation of the line tangent to f at x=0
b. Show that g has the same tangent line as f at x=0

2. Given x^2 + 3y^2 = 1
a. Find y'
b. Find the slope of the curve at the point (0.5, 0.5)
c. Find the equation of the line tangent to the ellipse at (0.5, 0.5)

Thanks a lot

 

[arthur ohlsten]

f[x]=-x^3-2x^2+x+1 let x=0 y=?
at x=0 y= 1

derivative of x^n = n x^(n-1)
derivative of a constant=0
a) f ' [x] = -3x^2-4x+1
at x=0 f'[0]=1 this is the slope of a tangent line at x=0

y=mx+b but m=1
y= 1x+b at x=0 y=1 find b
1=0+b b=1

tangent line y=x+1

b) g[x]=ln[x+1]+1
at x=0 y= ln[0+1] +1
at x=0 y=1

derivative of ln u = 1/u du/dx

g'[x]= 1/[x+1]
at x=0 g'[0]=1 this is slope at x=0 y=1 of g[x]
-------------------------------------------------------------------------------------
2)
x^2+3y^2=1

a) take derivative with respect to x
2x+6y dy/dx =0 solve for dy/dx
dy/dx = -x/3y answer

b) at .5,.5 what is dy/dx? substitute
dy/dx = -.5/1.5
dy/dx=-1/3 answer

c) at .5,.5 what is the tangent line?
y=mx+b but m=-1/3
y=[-1/3]x+b substitute .5,.5
.5= -[1/3].5 +b
b= [4/3].5
b=2/3

y=-1/3x +2/3 answer

Arthur