a function is continuous at a point if:
(1) the function is defined at the point
(2) the limit as x goes to the point exists
(3) the limit is equal to the value of the function
as for differentiable everywhere. note that a function is differentiable at a point if the limit exists. we then denote this limit by
Hint: if the function is defined piecewise about the point , we need to ensure
or, just find the derivative of the pieces and make sure it is continuous in much the same way as you made the original function continuous.
do you know what the domain and range are?2) a)I need to find the domain and range of the derivative of h(x) = x + sqrt[x] which i found to be 1 + .5x^(-.5).
I was never good at finding domain and range.
you are looking for a creature that looks like , this is what we call a straight line. is the slope, and is the y-intercept. you should be aware, that the derivative gives the formula for the slope at any value of x. thus, the slope of the tangent line at the point where x = 1 is . therefore, the slope of the normal line is .b) determine, in standard form, the equation of the line normal to the curve, y = h(x), at point (1, h(1)).
(in case anyone doesnt know, normal means perpendicular to the tangent line)
thus, you are looking for a line with slope passing through the point . i suppose you can find the equation of a line given this information.
the standard form of a line is where are constants