Which is greater?

Question

Which is greater: the instantaneous rate of change of f(x)= \sqrt(3x+4) at x=0 or at x=4? How do you know?

Answer 1

f =y= sqr(3x+4)

y^2 = 3x+4

3x= y^2 -4

x = (1/3)y^2 -4/3 is a horizontal parabola with vertex = (-4/3,0)

or rather it's the top half of a parabola

take the derivative with respect to x

(2/3)yy' = 1

y'=dy/dx = slope = instantaneous rate of change =

3/2y= 3/2(sqr(3x+4))

y'(0) = 3/2sqr4= 3/2(2)= 3/4= .75

y'(4) = 3/2sqr16 = 3/2(4) = 3/8 = .375

y'(0)>y'(4)

y'(0) is twice as large as y'(4)

it might help to graph the half parabola

then look at the points on the parabola corresponding to x=0 and x=4

f(0) = sqr(4)=2, f(4) = sqr16 = 4

that's the points (0,2) and (4,4)

the point where a tangent line is steeper means the derivative is larger there

f'(x) is the slope of the tangent line

Answer 2

Which is greater?

f'(0) or f'(4)