consider the function shown in the figure below.\n(a) write an expression involving the values of $x$ and $y…

consider the function shown in the figure below.\n(a) write an expression involving the values of $x$ and $y = f(x)$ marked on the graph that gives the slope of the line joining the endpoints of the graph.\nslope =\n(b) on a print - out of this problem, draw the tangent line to the curve at the point $(x2,y2)$. how does the slope of this line compare with that of your line in (a)?\nslope of this tangent line? slope of line from (a)\n(c) give another point on the curve where the slope of the line is approximately equal to the slope of your tangent line in (b).\nx =

consider the function shown in the figure below.\n(a) write an expression involving the values of $x$ and $y = f(x)$ marked on the graph that gives the slope of the line joining the endpoints of the graph.\nslope =\n(b) on a print - out of this problem, draw the tangent line to the curve at the point $(x2,y2)$. how does the slope of this line compare with that of your line in (a)?\nslope of this tangent line? slope of line from (a)\n(c) give another point on the curve where the slope of the line is approximately equal to the slope of your tangent line in (b).\nx =

Answer

Explanation:

Step1: Recall slope - formula

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. The endpoints of the graph are $(x_0,y_0)$ and $(x_6,y_6)$. So the slope of the line joining the endpoints is $\frac{y_6 - y_0}{x_6 - x_0}$.

Step2: Analyze the slope of the tangent line

The function is decreasing. The tangent line at $(x_2,y_2)$ has a steeper negative - slope compared to the line joining the endpoints (since the line joining the endpoints is a secant line that "averages" the change over a larger interval). So the slope of the tangent line at $(x_2,y_2)$ is less than the slope of the line in (a).

Step3: Find a point with similar slope

Visually, a point on the curve with a similar slope to the tangent line at $(x_2,y_2)$ is around $x = x_4$.

Answer:

(a) $\frac{y_6 - y_0}{x_6 - x_0}$ (b) $<$ (c) $x_4$