For what value of #c# is the line #y=2x + c# a tangent to the parabola #y=x^2-x-2# ?
For what value of #c# is the line #y=2x + c# a tangent to the parabola #y=x^2-x-2# ?
↳Redirected from
"A 5.00 L sample of helium at STP expands to 15.0 L. What is the new pressure on the gas?"
This is an easy calculus problem but it's currently classified as Algebra so let's do it using Algebra I.
#f(x) = x^2 - x - 2 #
#f(x) = f(r + (x-r)) = (r+(x-r))^2 - (r+(x-r)) - 2 #
# f(x) = r^2 + 2r (x-r) + (x-r)^2 - r - (x-r) - 2 #
#f(x) = (r^2 - r - 2) + (2r-1)(x-r) + (x-r)^2 #
When #x# is near #r# then #x-r# is small and #(x-r)^2# is smaller still. So the best linear approximation to #f# near #x=r# is gotten by dropping the #(x-r)^2# term. That's the tangent line:
#y = (r^2 - r - 2) + (2r-1) (x-r) #
#y = (2r-1)x -r^2-2 #
Comparing that to #y=2x+c # we get
#2r - 1 = 2 #
# r = 3/2 #
#c = -r^2 - 2 = - 9/4 - 2 = - 17/4 #
Check: Plot (2x - 17/4 - y)(x^2 - x - 2 - y) = 0
graph{(2x - 17/4 - y)(x^2 - x - 2 - y) = 0 [-3, 5, -3, 2]}
