Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

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Step 1: Evaluate the position function at the given points. Calculate s(0.5) by substituting t = 0.5 into the function s(t) = -16t^2 + 100t. Similarly, calculate s(2) by substituting t = 2 into the function.

Step 2: Determine the coordinates of the points on the graph. Use the results from Step 1 to find the coordinates (0.5, s(0.5)) and (2, s(2)).

Step 3: Calculate the slope of the secant line. Use the formula for the slope of a line, m = (y2 - y1) / (x2 - x1), where (x1, y1) = (0.5, s(0.5)) and (x2, y2) = (2, s(2)).

Step 4: Sketch the graph of the position function s(t) = -16t^2 + 100t. Plot the points (0.5, s(0.5)) and (2, s(2)) on the graph and draw the secant line passing through these points.

Step 5: Interpret the slope of the secant line. Explain that the slope represents the average velocity of the object over the interval from t = 0.5 to t = 2. Discuss how this average velocity relates to the object's motion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Function

The position function s(t) describes the location of an object at any given time t. In this case, s(t) = -16t^2 + 100t represents the vertical motion of an object under the influence of gravity, where the coefficients indicate the effects of gravitational acceleration and initial velocity.

Secant Line

A secant line is a straight line that intersects a curve at two or more points. In this context, the secant line connects the points (0.5, s(0.5)) and (2, s(2)), providing a way to approximate the average rate of change of the position function over the interval from t = 0.5 to t = 2.

Slope of the Secant Line

The slope of the secant line is calculated as the change in position divided by the change in time, represented mathematically as (s(2) - s(0.5)) / (2 - 0.5). This slope indicates the average velocity of the object over the specified time interval, reflecting how quickly the object's position changes as it moves.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).