To identify the coordinates of four points on the function y=1/x +x where the slope of the tangent line at that point is zero, undefined, positive, and negative, we first need to take the derivative of the function. The derivative of y=1/x +x is y'=-1/x^2 +1. We can set y'=0 to find the x-coordinate of the point where the slope of the tangent line is zero. Solving for x, we get x=1. To find the y-coordinate, we substitute x=1 into the original function and get y=2. Therefore, the point where the slope of the tangent line is zero is (1, 2).
To find the x-coordinate of the point where the slope of the tangent line is undefined, we need to look for values of x that make the denominator of the derivative equal to zero. In this case, the denominator is x^2, so the slope of the tangent line is undefined when x=0. Therefore, the point where the slope of the tangent line is undefined is (0, 0).
To find the x-coordinates of the points where the slope of the tangent line is positive or negative, we can look at the sign of the derivative on either side of the point x=1. We know that the slope of the tangent line is positive when the derivative is greater than zero, and negative when the derivative is less than zero. We can evaluate the derivative at x=0.5 and x=1.5 to find the sign of the derivative on either side of x=1. At x=0.5, y'=-3, so the slope of the tangent line is negative. At x=1.5, y'=0.44, so the slope of the tangent line is positive. Therefore, the points where the slope of the tangent line is negative and positive are (0.5, 2.5) and (1.5, 1.83), respectively.
To evaluate the derivative at the points we identified, we substitute each x-coordinate into the derivative y'=-1/x^2 +1. The derivative at x=0 is undefined, at x=0.5 it is negative, at x=1 it is zero, and at x=1.5 it is positive.
In summary, the continuity of a function is related to the limit of the function as x approaches a certain value. If the limit exists and is equal to the value of the function at that point, then the function is continuous at that point. The slope of the tangent line at a point on a function is related to the derivative of the function at that point. A positive derivative indicates that the function is increasing at that point, a negative derivative indicates that the function is decreasing, and a zero derivative indicates that the function has a horizontal tangent line at that point. The derivative can also be used to find the instantaneous rate of change of the function at a point. If a function is differentiable at a point, then it is also continuous at that point. However, the converse is not always true, as a function can be continuous but not differentiable at a point where the function has a sharp corner or a vertical tangent line.