Questions: a. What is the slope of the tangent line to the graph at x=1.5 ? slope = b. What is the slope of the tangent line to the graph at x=-1 ? slope = -2 c. What is the slope of the secant line that goes through the points with x-coordinates x=0.5 and x=1 ? slope =

Solution Steps

The equation of the graph is $y = x^2$. The derivative of this function is $y' = 2x$. The derivative gives the slope of the tangent line at any x value.

Substitute $x = 1.5$ into the derivative $y' = 2x$: $y' = 2(1.5) = 3$.

Substitute $x = -1$ into the derivative $y' = 2x$: $y' = 2(-1) = -2$

The secant line goes through the points with x-coordinates x = 0.5 and x = 1.

First, find the corresponding y-coordinates using the equation $y = x^2$. When $x=0.5$, $y = (0.5)^2 = 0.25$. When $x=1$, $y = (1)^2 = 1$.

Thus the two points are $(0.5, 0.25)$ and $(1, 1)$.

The slope of the secant line is given by the change in y divided by the change in x: $m = \frac{1 - 0.25}{1 - 0.5} = \frac{0.75}{0.5} = 1.5$.

Final Answer: