Graphs dealing with integrals

[paulxzt]

Can someone help me with these, I have absolute no clue how to start.


Let L be the line tangent to the graph of y = x^n at the point (1,1) where n>1

a) Find the integral of 0 to 1 of x^n in terms of n: (Is this right?)

( x^n+1 ) / (n + 1) evaluated from 0 to 1 = 1/(n+1)

b) Let T be the triangular region bounded by L, the x-axis and the line x=1. Show that the area of T is 1/2n.

c)Let S be the region bounded by the graph of y = x^n, the line L, and the x-axis. Express the area of S in terms of n and determine the value of n that maximizes the area of S.

Any help is appreciated.

 

[skeeter]

Can someone help me with these, I have absolute no clue how to start.


Let L be the line tangent to the graph of y = x^n at the point (1,1) where n>1

a) Find the integral of 0 to 1 of x^n in terms of n: (Is this right?)

( x^n+1 ) / (n + 1) evaluated from 0 to 1 = 1/(n+1)

correct

b) Let T be the triangular region bounded by L, the x-axis and the line x=1. Show that the area of T is 1/2n.

y = xⁿ
dy/dx = nx^n-1
at x = 1, dy/dx = n
the tangent line L is y - 1 = n(x - 1) or y = nx - n + 1
set y = 0 to find the x-intercept ...
nx - n + 1 = 0
nx = n - 1
x = (n-1)/n

distance from the x-intercept to x = 1 (the base of the triangle) is ...
1 - (n-1)/n = 1/n

area = (1/2)bh = (1/2)(1/n)(1) = 1/(2n)


c)Let S be the region bounded by the graph of y = x^n, the line L, and the x-axis. Express the area of S in terms of n and determine the value of n that maximizes the area of S.

S = 1/(n+1) - 1/(2n)
find dS/dn and maximize

Any help is appreciated.