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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.65a
Chapter 6, Problem 6.1.65a

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.
Theo: vT(t)=10, for t≥0
Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


a. Graph the velocity function for both riders. 

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Identify the velocity functions for both riders: Theo's velocity is constant, given by \(v_T(t) = 10\) for \(t \geq 0\), and Sasha's velocity is piecewise, given by \(v_S(t) = 15t\) for \(0 \leq t \leq 1\) and \(v_S(t) = 15\) for \(t > 1\).
Set up the time axis \(t\) starting from 0 and extending beyond 1 hour to capture both parts of Sasha's velocity function.
For Theo's velocity, draw a horizontal line at \(v = 10\) since his velocity is constant for all \(t \geq 0\).
For Sasha's velocity, first draw a line starting at the origin \((0,0)\) and increasing linearly to the point \((1, 15)\) because \(v_S(t) = 15t\) is a linear function on \(0 \leq t \leq 1\).
Then, for \(t > 1\), draw a horizontal line at \(v = 15\) to represent Sasha's constant velocity after 1 hour.

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Key Concepts

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

Velocity as a Function of Time

Velocity describes the rate of change of position with respect to time. In this problem, velocity functions vT(t) and vS(t) give the instantaneous speed of Theo and Sasha at any time t, which is essential for plotting their motion on a graph.
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Using The Velocity Function

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. Sasha's velocity vS(t) changes definition at t=1, requiring careful graphing of each segment separately to accurately represent her velocity over time.
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Piecewise Functions

Graphing Functions

Graphing involves plotting points and connecting them to visualize how a function behaves over its domain. Understanding how to graph constant and linear velocity functions helps illustrate the riders' speeds and compare their motion visually.
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Graph of Sine and Cosine Function
Related Practice
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