A GPS device tracks the elevation EEE (in feet) of a hiker walking in the mountains. The elevation ttt hours after beginning the hike is given in the figure. Find the slope of the secant line that passes through points AAA and BBB. Interpret your answer as an average rate of change over the interval 1≤t≤31\leq{t}\leq{3}1≤t≤3.

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

Problem 75a

Chapter 1, Problem 75a

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>
Find the slope of the secant line that passes through points $A$ and $B$ . Interpret your answer as an average rate of change over the interval $1$\leq{t}\]\leq{3}$$ .

Verified step by step guidance

Identify the coordinates of points A and B from the given figure. Let's assume point A corresponds to t = 1 and point B corresponds to t = 3. The coordinates will be (1, E(1)) and (3, E(3)), where E(t) is the elevation at time t.

Recall that the slope of a secant line through two points (x1, y1) and (x2, y2) is given by the formula: $ m = \frac{y2 - y1}{x2 - x1} $.

Substitute the coordinates of points A and B into the slope formula: $ m = \frac{E(3) - E(1)}{3 - 1} $.

Simplify the expression: $ m = \frac{E(3) - E(1)}{2} $. This represents the average rate of change of elevation over the interval from t = 1 to t = 3.

Interpret the result: The slope of the secant line, $ m $, represents the average rate of change of the hiker's elevation in feet per hour over the time interval from 1 to 3 hours.

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Key Concepts

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

Slope of a Secant Line

The slope of a secant line between two points on a graph represents the average rate of change of the function over the interval defined by those points. It is calculated using the formula (E(B) - E(A)) / (t(B) - t(A)), where E represents elevation and t represents time. This concept is crucial for understanding how the elevation changes as the hiker progresses over time.

Average Rate of Change

The average rate of change of a function over an interval gives a measure of how much the function's output changes per unit of input over that interval. In this context, it quantifies how the hiker's elevation changes on average for each hour of hiking between the specified times. This concept helps in interpreting the slope of the secant line in practical terms.

Function Representation

In this problem, the elevation E as a function of time t is represented graphically. Understanding how to read and interpret this graph is essential for identifying the points A and B, which are necessary for calculating the slope of the secant line. This concept emphasizes the relationship between graphical representations and mathematical calculations in calculus.

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